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PROFESSOR: Now
today, we're going
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00:00:24,270 --> 00:00:26,740
to talk about random walks.
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00:00:26,740 --> 00:00:29,940
And in particular, we're going
to look at a classic phenomenon
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00:00:29,940 --> 00:00:32,750
known as Gamblers Ruin.
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00:00:32,750 --> 00:00:34,760
It's a great way
to end the term,
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because the solution requires
several of the techniques
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that we've developed
since the midterm.
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00:00:40,740 --> 00:00:42,500
So it's actually a good review.
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00:00:42,500 --> 00:00:43,730
We'll review recurrences.
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00:00:43,730 --> 00:00:46,120
We'll review a lot
of probability laws.
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00:00:46,120 --> 00:00:49,050
And it's actually a
nice problem to look at.
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00:00:49,050 --> 00:00:52,090
It's another example where you
get a non-intuitive solution
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00:00:52,090 --> 00:00:53,740
using probability.
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00:00:53,740 --> 00:00:56,560
And if you like to
gamble, it's really good
22
00:00:56,560 --> 00:00:59,330
that you look at this problem
before you go to Vegas or down
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00:00:59,330 --> 00:01:01,050
to Foxwoods.
24
00:01:01,050 --> 00:01:07,165
Now the Gambler's Ruin problem,
you start with n dollars.
25
00:01:19,330 --> 00:01:22,390
And we're going to do a
simplified version, where
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00:01:22,390 --> 00:01:27,080
in each bet, you win
$1 or you lose $1.
27
00:01:27,080 --> 00:01:29,680
Now, these days, there are not
many bets in a casino for $1.
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00:01:29,680 --> 00:01:31,150
It's more like $10.
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00:01:31,150 --> 00:01:34,250
But just to make it
simple for counting,
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00:01:34,250 --> 00:01:37,980
we're going to
assume that each bet
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00:01:37,980 --> 00:01:46,720
you win $1 with
probability p, and you lose
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00:01:46,720 --> 00:01:53,900
$1 with probability 1 minus p.
33
00:01:53,900 --> 00:01:57,120
And in this version,
we're going to assume
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00:01:57,120 --> 00:02:00,610
you keep playing until
one of two things
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00:02:00,610 --> 00:02:04,380
happens-- you get
ahead by m dollars,
36
00:02:04,380 --> 00:02:10,530
or you lose all the money you
came with-- all n dollars.
37
00:02:10,530 --> 00:02:19,850
So you play until you
win m more-- net m plus--
38
00:02:19,850 --> 00:02:23,780
or you lose n.
39
00:02:23,780 --> 00:02:25,030
And that's where you go broke.
40
00:02:25,030 --> 00:02:25,904
You run out of money.
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00:02:28,335 --> 00:02:30,460
And we're going to assume
you don't borrow anything
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00:02:30,460 --> 00:02:31,085
from the house.
43
00:02:33,750 --> 00:02:36,610
All right, and we're going
to look at the probability
44
00:02:36,610 --> 00:02:39,870
that you come out a winner
versus going home broke--
45
00:02:39,870 --> 00:02:42,690
that you made m dollars.
46
00:02:42,690 --> 00:02:46,510
Now, the game we're going
to analyze is roulette,
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00:02:46,510 --> 00:02:48,330
but the technique
works for any of them.
48
00:02:54,160 --> 00:02:55,710
How many people
have played roulette
49
00:02:55,710 --> 00:02:57,850
before in some form or another?
50
00:02:57,850 --> 00:03:00,200
OK, so this is a
game where there's
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00:03:00,200 --> 00:03:03,310
the ball that goes around the
dish, and you spin the wheel.
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00:03:03,310 --> 00:03:07,170
And there's 36
numbers from 1 to 36.
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00:03:07,170 --> 00:03:09,260
Half of them are
red, half are black.
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00:03:09,260 --> 00:03:12,727
And then there's the zero and
the double zero that are green.
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00:03:12,727 --> 00:03:14,310
And we're going to
look at the version
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00:03:14,310 --> 00:03:17,000
where you just bet
on red or black.
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00:03:17,000 --> 00:03:19,710
And you win if the ball
lands on a slot that's red.
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00:03:19,710 --> 00:03:21,330
And there's 18 of those.
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00:03:21,330 --> 00:03:23,690
And you lose otherwise.
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00:03:23,690 --> 00:03:28,530
So in this case, the
probability of winning, p,
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00:03:28,530 --> 00:03:31,520
is there's 18 chances to win.
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00:03:31,520 --> 00:03:33,760
And it's not 36 total.
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00:03:33,760 --> 00:03:38,200
It's 38 total because of the
zero and the double zero.
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00:03:38,200 --> 00:03:42,680
All right so this is
9/19 chance of winning
65
00:03:42,680 --> 00:03:45,960
and a 10/19 chance of losing.
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00:03:45,960 --> 00:03:51,400
And so this is a game that has a
chance of winning of about 47%,
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00:03:51,400 --> 00:03:53,370
so it's almost a fair game.
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00:03:53,370 --> 00:03:54,869
It's not 50-50.
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00:03:54,869 --> 00:03:57,160
And that's because the casino's
got to make some money.
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00:03:57,160 --> 00:03:59,400
I mean, they have
the big facility.
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00:03:59,400 --> 00:04:02,010
They're giving you free
drinks, and all the rest.
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00:04:02,010 --> 00:04:03,880
So they got to
make money somehow.
73
00:04:03,880 --> 00:04:07,440
And they make money on
this bet because they're
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00:04:07,440 --> 00:04:12,430
going to make $0.03
on the dollar here.
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00:04:12,430 --> 00:04:13,780
You're going to wager.
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00:04:13,780 --> 00:04:18,230
And then you're going
to come back with 47%.
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00:04:18,230 --> 00:04:20,339
And people generally
are fine with that.
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00:04:20,339 --> 00:04:22,900
They don't expect to have
the odds in their favor
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00:04:22,900 --> 00:04:26,010
when you're gambling
in a casino.
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00:04:26,010 --> 00:04:30,180
Now, in an effort to sort
of come home a winner,
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00:04:30,180 --> 00:04:32,780
the way people do that--
knowing that the odds are
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00:04:32,780 --> 00:04:35,980
a little against
them-- is they might
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00:04:35,980 --> 00:04:38,580
put more money in
their pocket coming in
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00:04:38,580 --> 00:04:40,050
than they expect to win.
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00:04:40,050 --> 00:04:43,760
So often, you'll see
people come into the casino
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00:04:43,760 --> 00:04:48,829
with the goal of winning 100,
but they start with 1,000
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00:04:48,829 --> 00:04:49,495
in their pocket.
88
00:04:52,730 --> 00:04:55,440
So they're willing
to risk $1,000,
89
00:04:55,440 --> 00:04:59,790
but they're going to quit
happy if they get up 100.
90
00:04:59,790 --> 00:05:03,450
OK so you either go
home with $1,100,
91
00:05:03,450 --> 00:05:05,550
or you're going home
with $0, in this case.
92
00:05:05,550 --> 00:05:07,760
And you came with $1,000.
93
00:05:07,760 --> 00:05:10,300
And this means that you're--
at least the thinking goes--
94
00:05:10,300 --> 00:05:13,890
this means you're more
likely to go home happy.
95
00:05:13,890 --> 00:05:16,110
If you quit when
you get up by 100,
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00:05:16,110 --> 00:05:17,580
you're more likely
to land there,
97
00:05:17,580 --> 00:05:21,650
because it's almost a fair game,
than you are to lose all 1,000.
98
00:05:21,650 --> 00:05:23,570
That's the thinking anyway.
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00:05:23,570 --> 00:05:27,090
In fact, my mother-in-law
plays roulette, red and black,
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00:05:27,090 --> 00:05:30,040
and she follows the strategy.
101
00:05:30,040 --> 00:05:32,570
And she claims that she
does this for that reason--
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00:05:32,570 --> 00:05:35,510
that she almost always wins.
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00:05:35,510 --> 00:05:37,820
She goes home happy
almost always.
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00:05:37,820 --> 00:05:40,290
And that's the
important thing here.
105
00:05:40,290 --> 00:05:42,500
And it does reasonable,
because after all, roulette
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00:05:42,500 --> 00:05:45,980
is almost a fair game.
107
00:05:45,980 --> 00:05:47,470
So what do you think?
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00:05:47,470 --> 00:05:49,100
How many people
think she's right
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00:05:49,100 --> 00:05:52,650
that she almost always wins?
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00:05:52,650 --> 00:05:53,700
Anybody?
111
00:05:53,700 --> 00:05:54,900
I have sort of set it up.
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00:05:54,900 --> 00:05:56,775
It's my mother-in-law,
after all,
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00:05:56,775 --> 00:05:58,380
so probably she's
going to be wrong.
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00:06:01,740 --> 00:06:06,480
Well, how many people think
it's better than a 50% chance
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00:06:06,480 --> 00:06:11,610
you win $100 before
you lose $1,000?
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00:06:11,610 --> 00:06:15,030
That's probably
more-- how many people
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00:06:15,030 --> 00:06:19,350
think you're more likely to
lose $1,000 before you win $100?
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00:06:19,350 --> 00:06:21,960
Wow, OK, so you've been
to 6.04 too long now.
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00:06:21,960 --> 00:06:24,420
OK, what about this-- how
many people think you're
120
00:06:24,420 --> 00:06:30,505
more likely to lose
$10,000 than to win $100?
121
00:06:30,505 --> 00:06:32,130
All right, how many
people think you're
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00:06:32,130 --> 00:06:33,500
more likely to lose $1 million?
123
00:06:36,697 --> 00:06:38,030
A bunch of you still think that.
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00:06:38,030 --> 00:06:40,370
OK, well, you're right.
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00:06:40,370 --> 00:06:45,830
In fact, it is almost
certain you will go broke,
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00:06:45,830 --> 00:06:51,730
no matter how much money you
bring, before you win $100.
127
00:06:51,730 --> 00:06:55,290
In fact, we're
going to prove today
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00:06:55,290 --> 00:07:02,080
that the probability that
you win $100 before losing
129
00:07:02,080 --> 00:07:04,870
$100 million if you
stayed long enough-- that
130
00:07:04,870 --> 00:07:09,930
takes a while-- the chance you
go home a winner is less than 1
131
00:07:09,930 --> 00:07:14,470
in 37,648.
132
00:07:14,470 --> 00:07:18,326
You have no chance
to go home happy.
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00:07:18,326 --> 00:07:20,950
So my mother-in-law's telling me
the story about how she always
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00:07:20,950 --> 00:07:21,740
goes home happy.
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00:07:21,740 --> 00:07:24,960
And I'm saying, no, no,
wait a minute, you can't.
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00:07:24,960 --> 00:07:26,230
You never went home happy.
137
00:07:26,230 --> 00:07:28,040
Let's be honest.
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00:07:28,040 --> 00:07:29,057
It can't be.
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00:07:29,057 --> 00:07:30,390
She goes, no, no, no, it's true.
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00:07:30,390 --> 00:07:32,905
I go, no, look, there's
a mathematical proof.
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00:07:32,905 --> 00:07:33,530
I have a proof.
142
00:07:33,530 --> 00:07:38,140
I can show you my proof-- very
unlikely you go home a winner.
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00:07:38,140 --> 00:07:40,150
So somehow, she's
not very impressed
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00:07:40,150 --> 00:07:42,300
with the mathematical proof.
145
00:07:42,300 --> 00:07:43,320
And she keeps insisting.
146
00:07:43,320 --> 00:07:45,400
And I keep trying to
show her the proof.
147
00:07:45,400 --> 00:07:48,170
And anyway, I hope I'll have
more luck with you guys today
148
00:07:48,170 --> 00:07:51,520
in showing you the proof that
the chance you go home happy
149
00:07:51,520 --> 00:07:54,600
here is very, very small.
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00:07:54,600 --> 00:07:56,240
Now, in the end, I
didn't convince her,
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00:07:56,240 --> 00:07:59,680
but we'll see how
we do here today.
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00:07:59,680 --> 00:08:02,940
Now, in order to see why this
probability is so stunningly
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00:08:02,940 --> 00:08:06,690
small-- you would just
never guess it's that low--
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00:08:06,690 --> 00:08:09,510
we've got to learn
about random walks.
155
00:08:09,510 --> 00:08:12,010
And they come up in all
sorts of applications.
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00:08:12,010 --> 00:08:16,330
In fact, page rank--
that got Google started--
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00:08:16,330 --> 00:08:19,390
it's all based on a random
walk through the Web
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00:08:19,390 --> 00:08:24,600
or through the links on
web pages on that graph.
159
00:08:24,600 --> 00:08:26,380
Now, for the gambling
problem, we're
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00:08:26,380 --> 00:08:28,180
going to look at a
very special case--
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00:08:28,180 --> 00:08:30,210
probably the simplest
case of a random walk--
162
00:08:30,210 --> 00:08:33,390
and that's a
one-dimensional random walk.
163
00:08:33,390 --> 00:08:35,049
In a one-dimensional
random walk,
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00:08:35,049 --> 00:08:37,809
there's some value-- say
the number of dollars
165
00:08:37,809 --> 00:08:40,240
you've got in your pocket.
166
00:08:40,240 --> 00:08:43,309
And this value can
go up, or go down,
167
00:08:43,309 --> 00:08:47,310
or stay the same each time you
do something like make a bet.
168
00:08:47,310 --> 00:08:51,760
And each of this happens
with a certain probability.
169
00:08:51,760 --> 00:08:55,360
Now in this case, you
either go up by one,
170
00:08:55,360 --> 00:08:58,010
or you go down by one, and
you can't stay the same.
171
00:08:58,010 --> 00:09:00,100
Every bet you win
$1 or you lose $1.
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00:09:00,100 --> 00:09:02,490
So it's really a special case.
173
00:09:02,490 --> 00:09:05,700
And we can diagram
it as follows.
174
00:09:05,700 --> 00:09:09,700
We can put time, or the
number of bets, on this axis.
175
00:09:12,900 --> 00:09:16,365
And we can put the number
of dollars on this axis.
176
00:09:20,120 --> 00:09:25,010
Now in this case, we
start with n dollars.
177
00:09:25,010 --> 00:09:28,990
And we might win the first
bet, so we go to n plus 1.
178
00:09:28,990 --> 00:09:35,240
We might lose a bet, might lose
again, could win the next one,
179
00:09:35,240 --> 00:09:39,770
lose, win, lose, lose.
180
00:09:39,770 --> 00:09:45,210
So this corresponds to a
string-- win, lose, lose, lose,
181
00:09:45,210 --> 00:09:53,290
lose, win, lose, win,
lose, lose, lose.
182
00:09:53,290 --> 00:09:57,180
And when we win, we go up $1.
183
00:09:57,180 --> 00:10:01,260
When we lose, we go down $1.
184
00:10:01,260 --> 00:10:03,260
And it's called
one-dimensional, because there's
185
00:10:03,260 --> 00:10:05,050
just one thing that's changing.
186
00:10:05,050 --> 00:10:07,510
You're going up and down there.
187
00:10:07,510 --> 00:10:11,420
Now, the probability
of going up is p.
188
00:10:11,420 --> 00:10:13,990
And that's no matter
what happened before.
189
00:10:13,990 --> 00:10:17,380
It's a memoryless
independent system.
190
00:10:17,380 --> 00:10:20,950
The probability you win your
i-th bet has nothing to do--
191
00:10:20,950 --> 00:10:23,380
is totally independent,
mutually independent-- of all
192
00:10:23,380 --> 00:10:26,900
the other bets that
took place before.
193
00:10:26,900 --> 00:10:28,770
So let's write that down.
194
00:10:32,410 --> 00:10:40,040
So the probability
of an up move is p.
195
00:10:40,040 --> 00:10:46,210
The probability of a
down move is 1 minus p.
196
00:10:46,210 --> 00:10:52,720
And these are mutually
independent of past moves.
197
00:10:56,840 --> 00:10:59,080
Now, when you have
a random walk where
198
00:10:59,080 --> 00:11:01,060
the moves are
mutually independent,
199
00:11:01,060 --> 00:11:02,739
it has a special name.
200
00:11:02,739 --> 00:11:03,780
It's called a martingale.
201
00:11:10,120 --> 00:11:12,070
All random walks
don't have to have
202
00:11:12,070 --> 00:11:13,539
mutually independent steps.
203
00:11:13,539 --> 00:11:15,330
Say you're looking
about winning and losing
204
00:11:15,330 --> 00:11:17,422
a baseball game in a series.
205
00:11:17,422 --> 00:11:19,630
We looked at a scenario
where, if you lost yesterday,
206
00:11:19,630 --> 00:11:22,310
you're feeling lousy,
more likely to lose today.
207
00:11:22,310 --> 00:11:24,140
Not true in the
gambling case here.
208
00:11:24,140 --> 00:11:25,261
It's mutually independent.
209
00:11:25,261 --> 00:11:26,760
And that's the only
case we're going
210
00:11:26,760 --> 00:11:29,290
to study for random walks.
211
00:11:29,290 --> 00:11:36,585
Now, if p is not 1/2, the random
walk is said to be biased.
212
00:11:43,720 --> 00:11:45,430
And that's what
happens in the casino.
213
00:11:45,430 --> 00:11:49,040
It's biased in
favor of the house.
214
00:11:49,040 --> 00:11:53,460
If p equals 1/2, then the
random walk is unbiased.
215
00:12:00,440 --> 00:12:03,360
Now, in this particular
case that we're looking at,
216
00:12:03,360 --> 00:12:07,070
we have boundaries
on the random walk.
217
00:12:07,070 --> 00:12:10,000
There's a boundary
at 0, because you
218
00:12:10,000 --> 00:12:12,220
go home broke if
you lost everything.
219
00:12:12,220 --> 00:12:15,770
If the random walk ever
hit $0, you're done.
220
00:12:15,770 --> 00:12:23,100
And we're also going to
put a boundary at n plus m.
221
00:12:26,220 --> 00:12:30,690
So I'm going to have
a boundary here.
222
00:12:30,690 --> 00:12:37,180
So that if I win m dollars here,
I stop and I go home happy.
223
00:12:37,180 --> 00:12:41,790
If the random walk ever
goes here, then I stop.
224
00:12:41,790 --> 00:12:45,970
Those are called boundary
conditions for the walk.
225
00:12:45,970 --> 00:12:49,800
And what we want to do is
analyze the probability
226
00:12:49,800 --> 00:12:52,940
that we hit that top
boundary before we
227
00:12:52,940 --> 00:12:55,630
hit the bottom boundary.
228
00:12:55,630 --> 00:12:58,000
So we're going to define
that event to be W star.
229
00:13:00,580 --> 00:13:10,410
W star is the event that the
random walk hits T, which
230
00:13:10,410 --> 00:13:17,680
is n plus m, before it hits 0.
231
00:13:21,240 --> 00:13:26,080
In other words, you go home
happy without going broke.
232
00:13:26,080 --> 00:13:32,020
Let's also define D to be the
number of dollars at the start.
233
00:13:32,020 --> 00:13:34,930
And this is just going
to be n in our case.
234
00:13:40,960 --> 00:13:48,300
We're interested in, call it
X sub n is the probability
235
00:13:48,300 --> 00:13:53,750
that we go home happy given
we started with n dollars.
236
00:13:56,330 --> 00:13:57,740
And that's a function of n.
237
00:13:57,740 --> 00:14:00,650
So we'll make a
variable called X n.
238
00:14:00,650 --> 00:14:02,710
And we want to know what
that probability is.
239
00:14:02,710 --> 00:14:04,293
And of course, the
more you come with,
240
00:14:04,293 --> 00:14:07,775
you'd think it's a
higher chance of winning
241
00:14:07,775 --> 00:14:09,150
the more you have
in your pocket,
242
00:14:09,150 --> 00:14:11,250
because you can play for more.
243
00:14:11,250 --> 00:14:14,570
So the goal is to
figure this out.
244
00:14:14,570 --> 00:14:18,050
Now to do this, we could
use the tree method.
245
00:14:18,050 --> 00:14:22,110
But it gets pretty complicated,
because the sample space
246
00:14:22,110 --> 00:14:26,390
is the sample space of
all one-loss sequences.
247
00:14:26,390 --> 00:14:28,130
And how big is
that sample space?
248
00:14:32,106 --> 00:14:33,100
AUDIENCE: Infinite.
249
00:14:33,100 --> 00:14:35,050
PROFESSOR: Infinite.
250
00:14:35,050 --> 00:14:36,977
I could play forever.
251
00:14:36,977 --> 00:14:39,560
All right, now it turns out the
probability of playing forever
252
00:14:39,560 --> 00:14:40,080
is 0.
253
00:14:40,080 --> 00:14:43,350
And we won't prove that, but
there are an infinite number
254
00:14:43,350 --> 00:14:44,150
of sample points.
255
00:14:44,150 --> 00:14:46,150
So doing the tree method
is a little complicated
256
00:14:46,150 --> 00:14:49,060
when it's infinite.
257
00:14:49,060 --> 00:14:52,280
So what we're going
to do is use some
258
00:14:52,280 --> 00:14:54,950
of the theorems we've proved
over the last few weeks
259
00:14:54,950 --> 00:14:57,835
and set up a recurrence
to find this probability.
260
00:15:01,100 --> 00:15:04,640
Now, I'm going to tell you
what the recurrence is,
261
00:15:04,640 --> 00:15:09,730
and then prove
that that's right.
262
00:15:09,730 --> 00:15:17,580
So I claim that X
n is 0 probability
263
00:15:17,580 --> 00:15:21,320
if we start with $0.
264
00:15:21,320 --> 00:15:25,730
It's 1 if we start
with T dollars.
265
00:15:25,730 --> 00:15:36,580
And it's p times X n minus
1 plus 1 minus p X n plus 1
266
00:15:36,580 --> 00:15:43,100
if we start with between
$0 and T dollars.
267
00:15:47,110 --> 00:15:49,450
All right, so that's
what I claim X n is.
268
00:15:49,450 --> 00:15:51,960
And it's, of course, a
recursion that I've set up here.
269
00:15:51,960 --> 00:15:53,650
So let's see why
that's the case.
270
00:15:58,230 --> 00:16:00,930
OK, so let's check the 0 case.
271
00:16:00,930 --> 00:16:07,530
X 0 is the probability
we go home a winner given
272
00:16:07,530 --> 00:16:09,210
we started with $0.
273
00:16:12,510 --> 00:16:13,475
Why is that 0?
274
00:16:16,974 --> 00:16:17,890
AUDIENCE: [INAUDIBLE].
275
00:16:17,890 --> 00:16:19,386
PROFESSOR: What's that?
276
00:16:19,386 --> 00:16:21,180
AUDIENCE: [INAUDIBLE].
277
00:16:21,180 --> 00:16:23,280
PROFESSOR: Yeah,
you started broke.
278
00:16:23,280 --> 00:16:25,520
You never get off the
ground, because you
279
00:16:25,520 --> 00:16:27,730
quit as soon as you have $0.
280
00:16:27,730 --> 00:16:35,020
So you have no chance to win,
because you're broke to start.
281
00:16:35,020 --> 00:16:41,670
Let's check the next case,
X T-- case n equals T--
282
00:16:41,670 --> 00:16:45,570
is the probability you
go home a winner given
283
00:16:45,570 --> 00:16:49,310
you started with T dollars.
284
00:16:49,310 --> 00:16:51,750
Why is that 1?
285
00:16:51,750 --> 00:16:55,147
Why is that certain, sort
of from the definition?
286
00:16:55,147 --> 00:16:56,580
AUDIENCE: [INAUDIBLE].
287
00:16:56,580 --> 00:16:58,230
PROFESSOR: You already
have your money.
288
00:16:58,230 --> 00:16:59,560
You already hit
the top boundary,
289
00:16:59,560 --> 00:17:00,643
because you started there.
290
00:17:00,643 --> 00:17:02,440
Remember, you quit
and you're happy.
291
00:17:02,440 --> 00:17:05,900
Go home happy if
you hit T dollars.
292
00:17:05,900 --> 00:17:08,697
All right, so you're
guaranteed to go home happy,
293
00:17:08,697 --> 00:17:10,030
because you never make any bets.
294
00:17:10,030 --> 00:17:13,530
You started with all the money
you needed to go home happy.
295
00:17:13,530 --> 00:17:18,490
Then we have the
interesting case,
296
00:17:18,490 --> 00:17:20,849
where you start with
between 0 and T dollars.
297
00:17:20,849 --> 00:17:23,990
And now you're going
to make some bets.
298
00:17:23,990 --> 00:17:26,310
And then X n is the
probability-- just
299
00:17:26,310 --> 00:17:30,590
the definition-- of going
home happy-- i.e. winning
300
00:17:30,590 --> 00:17:34,680
and having T dollars,
if you start with n.
301
00:17:34,680 --> 00:17:38,710
Now, there's two cases to
analyze this, based on what
302
00:17:38,710 --> 00:17:40,380
happens in the first bet.
303
00:17:40,380 --> 00:17:43,600
You could win it, or
you could lose it.
304
00:17:43,600 --> 00:17:46,240
And then we're going to recurse.
305
00:17:46,240 --> 00:17:54,840
So we're going to define
E to be the event that you
306
00:17:54,840 --> 00:17:57,340
win the first bet.
307
00:18:00,570 --> 00:18:07,520
And E bar is the event that
you lose the first bet.
308
00:18:12,080 --> 00:18:14,340
Now, by the theory of
total probability, which
309
00:18:14,340 --> 00:18:18,120
we did in recitation
maybe a couple weeks ago,
310
00:18:18,120 --> 00:18:21,970
we can rewrite this depending
on whether E happened
311
00:18:21,970 --> 00:18:25,080
or the complement of E happened.
312
00:18:25,080 --> 00:18:28,370
And you get that the
probability is simply
313
00:18:28,370 --> 00:18:31,990
the probability of
going home happy
314
00:18:31,990 --> 00:18:34,844
and winning the
first bet times--
315
00:18:34,844 --> 00:18:36,510
and I've got to put
the conditioning in.
316
00:18:36,510 --> 00:18:37,580
That doesn't go away.
317
00:18:44,080 --> 00:18:45,510
So I'm breaking into two cases.
318
00:18:45,510 --> 00:18:51,350
The first one is you win the
first bet given D equals n,
319
00:18:51,350 --> 00:18:59,470
And the case where you lose the
first bet, given D equals n.
320
00:19:02,070 --> 00:19:04,601
Any questions here?
321
00:19:04,601 --> 00:19:06,530
The probability of
going home happy
322
00:19:06,530 --> 00:19:09,660
given you start with n
dollars is the probability
323
00:19:09,660 --> 00:19:17,050
of going home happy and winning
the first bet given D equals
324
00:19:17,050 --> 00:19:19,465
n plus the probability
of going home happy
325
00:19:19,465 --> 00:19:22,800
and losing the first
bet given D equals n--
326
00:19:22,800 --> 00:19:25,630
just those are the two cases.
327
00:19:25,630 --> 00:19:29,290
Now I can use the definition
of conditional probability
328
00:19:29,290 --> 00:19:31,256
to rewrite these.
329
00:19:31,256 --> 00:19:34,350
This is the probability--
you've got two events--
330
00:19:34,350 --> 00:19:39,520
that the first one
happens given D equals n
331
00:19:39,520 --> 00:19:43,440
times the probability
the second one happens
332
00:19:43,440 --> 00:19:49,180
given that the first one
happened and D equals n.
333
00:19:49,180 --> 00:19:51,560
This is just the definition
of conditional probability,
334
00:19:51,560 --> 00:19:54,180
when I've got an
intersection of events here.
335
00:19:54,180 --> 00:19:55,870
The probability
of both happening
336
00:19:55,870 --> 00:19:57,760
is the probability of
the first happening
337
00:19:57,760 --> 00:19:59,884
times the probability of
the second happening given
338
00:19:59,884 --> 00:20:00,960
that the first happened.
339
00:20:00,960 --> 00:20:05,570
And of course, everything is
in this universe of D equals n.
340
00:20:05,570 --> 00:20:07,670
So I've used it in a
little different twist
341
00:20:07,670 --> 00:20:09,490
than we had it before.
342
00:20:09,490 --> 00:20:11,240
The same thing over
here-- this now
343
00:20:11,240 --> 00:20:17,140
is the probability of E
prime given D equals n
344
00:20:17,140 --> 00:20:20,390
times the probability of
W star-- winning, going
345
00:20:20,390 --> 00:20:29,420
home happy-- given that you lost
the first bet and D equals n.
346
00:20:29,420 --> 00:20:32,370
That's D equals n there.
347
00:20:32,370 --> 00:20:34,540
So it looks like it's
got more complicated,
348
00:20:34,540 --> 00:20:36,005
but now we can
start simplifying.
349
00:20:38,650 --> 00:20:40,610
What's the
probability of winning
350
00:20:40,610 --> 00:20:44,022
the first bet given that
you started with n dollars?
351
00:20:44,022 --> 00:20:45,390
AUDIENCE: p.
352
00:20:45,390 --> 00:20:47,565
PROFESSOR: p-- in
fact, does this
353
00:20:47,565 --> 00:20:49,690
have anything to do with
the probability of winning
354
00:20:49,690 --> 00:20:51,730
the first bet?
355
00:20:51,730 --> 00:20:53,870
No, this is just p.
356
00:20:59,610 --> 00:21:03,290
Now, what about this thing?
357
00:21:03,290 --> 00:21:13,620
I am conditioning on
winning the first bet given
358
00:21:13,620 --> 00:21:15,355
and I start with n dollars.
359
00:21:19,750 --> 00:21:22,930
What's another way of
expressing I won the first bet
360
00:21:22,930 --> 00:21:25,310
and I started with n dollars?
361
00:21:25,310 --> 00:21:26,087
Yeah?
362
00:21:26,087 --> 00:21:27,520
AUDIENCE: You have n plus $1.
363
00:21:27,520 --> 00:21:31,860
PROFESSOR: I now have n
plus $1 going forward.
364
00:21:31,860 --> 00:21:34,900
And because I have a
martingale, and everything
365
00:21:34,900 --> 00:21:37,340
is mutually independent,
it's like the world
366
00:21:37,340 --> 00:21:38,700
starts all over again.
367
00:21:38,700 --> 00:21:41,440
I'm now in a state
with n plus $1,
368
00:21:41,440 --> 00:21:44,110
and I want to know
the probability
369
00:21:44,110 --> 00:21:45,375
that I go home happy.
370
00:21:45,375 --> 00:21:49,000
It doesn't matter how
I got the n plus $1.
371
00:21:49,000 --> 00:21:52,000
It's just going forward-- I
got n plus $1 in my pocket,
372
00:21:52,000 --> 00:21:54,270
I want to know the probability
of going home happy.
373
00:21:54,270 --> 00:21:57,040
So I reset to D equals n plus 1.
374
00:21:59,740 --> 00:22:03,892
So I replace this with that,
because however long it
375
00:22:03,892 --> 00:22:05,850
took me to get there and
all that stuff doesn't
376
00:22:05,850 --> 00:22:07,660
matter for this analysis.
377
00:22:07,660 --> 00:22:08,990
It's all mutually dependent.
378
00:22:13,110 --> 00:22:16,360
Probability of losing the
first bet given that I started
379
00:22:16,360 --> 00:22:20,490
with n dollars-- 1 minus p.
380
00:22:20,490 --> 00:22:23,290
Doesn't matter how
much I started with.
381
00:22:23,290 --> 00:22:29,120
And here, I want to know the
probability of going home happy
382
00:22:29,120 --> 00:22:31,950
given-- well, if I
lost the first bet
383
00:22:31,950 --> 00:22:36,080
and I started with
n, what have I got?
384
00:22:36,080 --> 00:22:37,087
n minus 1.
385
00:22:40,530 --> 00:22:44,746
It doesn't matter how
I got to n minus 1.
386
00:22:44,746 --> 00:22:46,370
Now this is going to
get really simple.
387
00:22:49,250 --> 00:22:52,720
What's another name
for that expression?
388
00:22:52,720 --> 00:22:53,830
X n plus 1.
389
00:22:57,560 --> 00:23:00,800
And another name
for this expression?
390
00:23:00,800 --> 00:23:02,003
X n minus 1.
391
00:23:05,680 --> 00:23:11,720
So we proved that X n equals
p X n plus 1 plus 1 minus
392
00:23:11,720 --> 00:23:12,770
p X n minus 1.
393
00:23:12,770 --> 00:23:15,100
And that's what I
claimed is true.
394
00:23:15,100 --> 00:23:18,380
So we finished the proof.
395
00:23:18,380 --> 00:23:20,214
Any questions?
396
00:23:20,214 --> 00:23:21,130
AUDIENCE: [INAUDIBLE].
397
00:23:21,130 --> 00:23:22,356
PROFESSOR: Did i screw it up?
398
00:23:22,356 --> 00:23:23,272
AUDIENCE: [INAUDIBLE].
399
00:23:23,272 --> 00:23:29,940
PROFESSOR: I claim probability
of winning-- so let's
400
00:23:29,940 --> 00:23:31,590
see if I have a wrong in here.
401
00:23:31,590 --> 00:23:32,950
I might have screwed it up.
402
00:23:32,950 --> 00:23:38,440
I think I proved
it's n plus 1, right?
403
00:23:38,440 --> 00:23:41,230
Yep, sure enough, I
think this is a plus 1.
404
00:23:41,230 --> 00:23:43,210
That's a minus 1.
405
00:23:43,210 --> 00:23:45,692
Now, it's always good to
check to you proved what you
406
00:23:45,692 --> 00:23:46,900
said you were going to prove.
407
00:23:46,900 --> 00:23:50,006
So I needed to change this.
408
00:23:50,006 --> 00:23:50,880
That's what I proved.
409
00:23:57,290 --> 00:23:58,239
Any other questions?
410
00:23:58,239 --> 00:23:59,780
That was a pretty
important question.
411
00:24:02,370 --> 00:24:07,620
All right, so we have
a recurrence for X n.
412
00:24:07,620 --> 00:24:10,430
Now, it's a little
funny looking at first,
413
00:24:10,430 --> 00:24:12,430
because normally
with a recurrence,
414
00:24:12,430 --> 00:24:15,770
X n would depend on X
sub i that are smaller--
415
00:24:15,770 --> 00:24:18,040
the i's are smaller than n.
416
00:24:18,040 --> 00:24:20,210
So it looks a little wacky.
417
00:24:20,210 --> 00:24:21,470
But is that a problem?
418
00:24:24,320 --> 00:24:26,550
I can just solve
for X n plus 1--
419
00:24:26,550 --> 00:24:29,550
just subtract this
and put it over there.
420
00:24:29,550 --> 00:24:30,540
So let's do that.
421
00:24:47,740 --> 00:24:53,820
OK, so if I solve for X n plus 1
up there, I'll put p X n plus 1
422
00:24:53,820 --> 00:25:03,210
on its own side, I get p X
n plus 1 minus X n plus 1
423
00:25:03,210 --> 00:25:08,690
minus p X n minus 1 equals 0.
424
00:25:08,690 --> 00:25:11,260
And I know that X 0 is 0.
425
00:25:11,260 --> 00:25:13,445
And I know that X T equals 1.
426
00:25:17,314 --> 00:25:18,855
Now, what type of
recurrence is this?
427
00:25:21,860 --> 00:25:22,770
AUDIENCE: Linear.
428
00:25:22,770 --> 00:25:26,130
PROFESSOR: Linear, good, so
it's a linear recurrence.
429
00:25:26,130 --> 00:25:28,853
And what type of linear
recurrence is it?
430
00:25:28,853 --> 00:25:29,800
AUDIENCE: Homogeneous.
431
00:25:29,800 --> 00:25:33,200
PROFESSOR: Homogeneous-- that's
the best case, simple case,
432
00:25:33,200 --> 00:25:34,039
that's good.
433
00:25:34,039 --> 00:25:35,830
The boundary conditions
are a little weird,
434
00:25:35,830 --> 00:25:38,820
because the recurrences
we all saw before,
435
00:25:38,820 --> 00:25:40,320
if we had two
boundary conditions it
436
00:25:40,320 --> 00:25:41,870
would be X0 and X1.
437
00:25:41,870 --> 00:25:43,260
Here it's X0 and X T.
438
00:25:43,260 --> 00:25:45,151
But all's you need are two.
439
00:25:45,151 --> 00:25:46,400
Doesn't matter where they are.
440
00:25:48,960 --> 00:25:50,220
So how do I solve that thing?
441
00:25:50,220 --> 00:25:54,070
What's the next thing I do?
442
00:25:54,070 --> 00:25:55,041
What is it?
443
00:25:55,041 --> 00:25:56,540
AUDIENCE: Characterize
the equation.
444
00:25:56,540 --> 00:25:57,560
PROFESSOR: Characterize
the equation.
445
00:25:57,560 --> 00:25:59,294
And what do you do
it that equation?
446
00:25:59,294 --> 00:26:00,210
AUDIENCE: [INAUDIBLE].
447
00:26:00,210 --> 00:26:01,668
PROFESSOR: Solve
it, get the roots.
448
00:26:01,668 --> 00:26:03,820
This'll be good
practice for the final,
449
00:26:03,820 --> 00:26:06,797
because you'll probably have
to do something like this.
450
00:26:06,797 --> 00:26:08,380
So that's the
characteristic equation.
451
00:26:12,800 --> 00:26:15,940
And what's the order of
this equation-- the degree?
452
00:26:20,340 --> 00:26:26,730
That's going to be 2, right?
453
00:26:26,730 --> 00:26:33,600
I'm going to have pr squared
minus r plus 1 minus p is 0.
454
00:26:33,600 --> 00:26:37,177
That's my
characteristic equation.
455
00:26:37,177 --> 00:26:37,760
Remember that?
456
00:26:37,760 --> 00:26:40,640
So I make this be
the constant term.
457
00:26:40,640 --> 00:26:42,420
Then I have the
first-order term, then
458
00:26:42,420 --> 00:26:45,210
the second-order term.
459
00:26:45,210 --> 00:26:48,570
All right, now I solve it.
460
00:26:48,570 --> 00:26:52,080
And that's easy for a
second-order equation.
461
00:26:52,080 --> 00:26:59,540
1 plus or minus the square
root of 1 minus 4p 1
462
00:26:59,540 --> 00:27:03,900
minus p over 2 p.
463
00:27:07,140 --> 00:27:08,030
Let's do that.
464
00:27:10,810 --> 00:27:15,380
OK, so this is 1 plus
or minus the square root
465
00:27:15,380 --> 00:27:20,460
of 1 minus 4p plus
4p squared over 2p.
466
00:27:28,689 --> 00:27:30,730
Just using the quadratic
formula and simplifying.
467
00:27:35,780 --> 00:27:38,930
And it works out really
nicely, because that
468
00:27:38,930 --> 00:27:42,410
is the square root of-- this
is just 1 minus 2p squared.
469
00:27:48,990 --> 00:27:55,740
So that's 1 plus or
minus 1 minus 2p over 2p.
470
00:27:55,740 --> 00:28:06,410
And that is 2 minus 2p over
2p or 1 minus 1 cancels,
471
00:28:06,410 --> 00:28:13,020
then minus 2p is 2p over 2p.
472
00:28:13,020 --> 00:28:17,590
So the answers, the roots
are divide by 2 on this one.
473
00:28:17,590 --> 00:28:22,947
I get 1 minus p over p and 1.
474
00:28:22,947 --> 00:28:23,780
Those are the roots.
475
00:28:27,582 --> 00:28:28,665
Are these roots different?
476
00:28:30,960 --> 00:28:32,460
Do I have the case
of a double root?
477
00:28:32,460 --> 00:28:36,407
Are the roots always different?
478
00:28:36,407 --> 00:28:37,490
They're usually different.
479
00:28:37,490 --> 00:28:39,880
What's the case where
these roots are the same?
480
00:28:39,880 --> 00:28:40,870
AUDIENCE: 0.5.
481
00:28:40,870 --> 00:28:43,760
PROFESSOR: 0.5, which is
sort of an interesting case
482
00:28:43,760 --> 00:28:45,990
in this game.
483
00:28:45,990 --> 00:28:49,110
Because if p equals 1/2, we
have an unbiased random walk.
484
00:28:49,110 --> 00:28:51,250
You got a fair game.
485
00:28:51,250 --> 00:28:53,730
And so it says right away,
well, maybe the result
486
00:28:53,730 --> 00:28:56,807
is going to be different for
a fair game than the game
487
00:28:56,807 --> 00:28:58,765
we're playing in the
casino, where it's biased.
488
00:29:01,300 --> 00:29:04,432
So let's look at the casino
game where p is not 1/2.
489
00:29:04,432 --> 00:29:05,640
Then the roots are different.
490
00:29:08,280 --> 00:29:10,000
Later, we'll go back
and analyze the case
491
00:29:10,000 --> 00:29:13,053
when the roots of the
same for the fair game.
492
00:29:20,310 --> 00:29:30,250
So if p is not 1/2, then
we can solve for X n.
493
00:29:30,250 --> 00:29:37,960
X n is some constant times the
first root to the nth power
494
00:29:37,960 --> 00:29:42,180
plus a constant times the
second root to the nth power.
495
00:29:42,180 --> 00:29:45,515
Remember, that's how it works
for any linear homogeneous
496
00:29:45,515 --> 00:29:46,015
recurrence.
497
00:29:49,300 --> 00:29:53,900
And that's easy, because
the second root was 1.
498
00:29:53,900 --> 00:29:57,200
This is just plus
B. 1 to the n is 1.
499
00:29:59,960 --> 00:30:03,050
How do I figure out
what A and B are?
500
00:30:03,050 --> 00:30:04,300
AUDIENCE: Boundary conditions.
501
00:30:04,300 --> 00:30:07,090
PROFESSOR: Boundary
conditions, very good.
502
00:30:07,090 --> 00:30:08,855
So let's look at the
boundary conditions.
503
00:30:30,250 --> 00:30:36,100
OK, so the first boundary
condition is at 0.
504
00:30:36,100 --> 00:30:40,410
So we have 0 equals X 0.
505
00:30:40,410 --> 00:30:44,570
Plugging in there-- oops
I forgot the n up here.
506
00:30:44,570 --> 00:30:48,140
Plugging in n equals 0--
well, this to the 0 is just 1.
507
00:30:48,140 --> 00:30:57,054
That is A plus B. That
means that B equals minus A.
508
00:30:57,054 --> 00:30:58,470
Then the second
boundary condition
509
00:30:58,470 --> 00:31:05,880
is 1 equals X sub T.
And that is A 1 minus
510
00:31:05,880 --> 00:31:13,442
p over p to the T plus
B, but B was minus A.
511
00:31:13,442 --> 00:31:16,860
And now I can solve for A.
512
00:31:16,860 --> 00:31:28,850
So that means that A equals
1 over 1 minus p over p
513
00:31:28,850 --> 00:31:32,200
to the T minus 1.
514
00:31:32,200 --> 00:31:38,610
And B is negative A--
minus 1 over 1 minus p,
515
00:31:38,610 --> 00:31:40,810
over p to the T minus 1.
516
00:31:43,540 --> 00:31:51,140
And then I plug those back
in to the formula for X n.
517
00:31:51,140 --> 00:31:54,320
So here's my constant A.
I multiply that times 1
518
00:31:54,320 --> 00:31:59,020
minus p over p to the
n, plus I add this in.
519
00:31:59,020 --> 00:32:02,620
So this means that the
probability of going home
520
00:32:02,620 --> 00:32:10,500
a winner is 1 minus p over p to
the n over that thing-- 1 minus
521
00:32:10,500 --> 00:32:14,830
p over p to the T
minus 1, plus the B
522
00:32:14,830 --> 00:32:22,910
term, which really is a minus
term here, is just minus 1.
523
00:32:22,910 --> 00:32:23,880
Put that on top here.
524
00:32:26,620 --> 00:32:30,350
That sort of looks
messy, but there's
525
00:32:30,350 --> 00:32:34,780
a simplification to get an
upper bound that's very close.
526
00:32:34,780 --> 00:32:38,870
In particular, if you have
a biased game against you--
527
00:32:38,870 --> 00:32:43,960
so if p is less than 1/2,
as it is in roulette,
528
00:32:43,960 --> 00:32:47,720
then this is a
number bigger than 1.
529
00:32:47,720 --> 00:32:53,105
That means that 1 minus p
over p is bigger than 1.
530
00:32:56,340 --> 00:32:57,990
So this is bigger than 1.
531
00:32:57,990 --> 00:32:58,990
This is bigger than 1.
532
00:32:58,990 --> 00:33:01,620
T is the upper limit.
533
00:33:01,620 --> 00:33:03,410
It's n plus m.
534
00:33:03,410 --> 00:33:05,970
So I've got a bigger number
down here than I do here.
535
00:33:05,970 --> 00:33:08,750
So overall, it's a
fraction less than 1.
536
00:33:08,750 --> 00:33:10,860
And when you have a
fraction less than 1,
537
00:33:10,860 --> 00:33:14,330
if you add 1 to the
numerator and denominator,
538
00:33:14,330 --> 00:33:15,480
it gets closer to 1.
539
00:33:15,480 --> 00:33:17,670
It gets bigger.
540
00:33:17,670 --> 00:33:23,460
So this is upper-bounded by
just adding 1 to each of these.
541
00:33:23,460 --> 00:33:28,440
Its upper-bounded by this
over that, which is 1 minus p
542
00:33:28,440 --> 00:33:36,860
over p to the n minus T.
And T is just n plus m.
543
00:33:36,860 --> 00:33:40,930
So this equals-- why don't
I turn it upside down?
544
00:33:40,930 --> 00:33:44,720
Make it p over 1 minus p to get
a fraction that's less than 1.
545
00:33:44,720 --> 00:33:50,950
T minus n, and that equals
p over 1 minus p to the m.
546
00:33:54,240 --> 00:33:56,987
And this is how much you're
trying to get ahead-- $100
547
00:33:56,987 --> 00:33:58,320
in the case of my mother-in-law.
548
00:34:01,230 --> 00:34:03,894
So what we've
proved-- let me state
549
00:34:03,894 --> 00:34:05,060
what we proved as a theorem.
550
00:34:24,929 --> 00:34:31,830
So we proved that if
p is less than 1/2--
551
00:34:31,830 --> 00:34:33,889
if you're more
likely to lose a bet
552
00:34:33,889 --> 00:34:40,420
than win it-- then
the probability
553
00:34:40,420 --> 00:34:51,150
that you win m dollars before
you lose n dollars is at most p
554
00:34:51,150 --> 00:34:55,510
over 1 minus p to the m.
555
00:34:55,510 --> 00:34:58,590
That's what we just proved.
556
00:34:58,590 --> 00:35:00,659
And so now you can plug
in values-- for example,
557
00:35:00,659 --> 00:35:01,200
for roulette.
558
00:35:05,260 --> 00:35:11,960
p equals 9/19, which
means that p over 1
559
00:35:11,960 --> 00:35:19,750
minus p-- that's going
to be 9/19 over 10/19,
560
00:35:19,750 --> 00:35:21,010
which is just 9/10.
561
00:35:24,740 --> 00:35:30,010
And if m-- the amount you
want to win-- is $100,
562
00:35:30,010 --> 00:35:33,165
and n is $1,000-- that's
what you start with
563
00:35:33,165 --> 00:35:36,420
and you're willing
to lose-- well,
564
00:35:36,420 --> 00:35:42,640
the probability you win--
you go home happy-- W
565
00:35:42,640 --> 00:35:50,310
star you win $100-- is
less than or equal to 9/10
566
00:35:50,310 --> 00:35:53,800
raised to the m, which is 100.
567
00:35:53,800 --> 00:35:57,610
So it's 9/10 of 100, and that
turns out to be less than 1
568
00:35:57,610 --> 00:36:06,280
in 37,648, which is where
that answer came from.
569
00:36:06,280 --> 00:36:07,800
Now you can see why
my mother-in-law
570
00:36:07,800 --> 00:36:11,640
may have got lost somewhere
here now in the calculations.
571
00:36:11,640 --> 00:36:13,560
But this is a proof
that the chance
572
00:36:13,560 --> 00:36:18,850
you win $100 before you lose
$1,000 is very, very small.
573
00:36:18,850 --> 00:36:22,810
Now, do you see why the answer
is no better than if you came
574
00:36:22,810 --> 00:36:26,310
with $1 million in your pocket?
575
00:36:26,310 --> 00:36:30,390
Say you came with n
equals $1 million.
576
00:36:30,390 --> 00:36:32,390
Why is the answer not changing?
577
00:36:35,690 --> 00:36:37,020
Yeah.
578
00:36:37,020 --> 00:36:39,930
AUDIENCE: Once you
lose, say, $1,000,
579
00:36:39,930 --> 00:36:42,650
you're already in
a really deep hole.
580
00:36:42,650 --> 00:36:44,130
PROFESSOR: That's the intuition.
581
00:36:44,130 --> 00:36:44,671
That's right.
582
00:36:44,671 --> 00:36:46,560
We're going to get
to that in a minute.
583
00:36:46,560 --> 00:36:49,590
I want to know from
the formula, why
584
00:36:49,590 --> 00:36:54,280
is it no difference if I come
with $1,000 versus $1 million?
585
00:36:54,280 --> 00:36:55,051
Yeah.
586
00:36:55,051 --> 00:36:56,592
AUDIENCE: The formula
doesn't have n.
587
00:36:56,592 --> 00:37:00,180
PROFESSOR: Yeah, the formula
has nothing to do with n.
588
00:37:00,180 --> 00:37:04,130
You could come with $100
trillion in your wallet,
589
00:37:04,130 --> 00:37:06,710
and it doesn't
improve this bound.
590
00:37:06,710 --> 00:37:10,260
This bound only depends on
what you're trying to win,
591
00:37:10,260 --> 00:37:11,530
not on how much you came with.
592
00:37:11,530 --> 00:37:13,890
So no matter how
much you come with,
593
00:37:13,890 --> 00:37:17,250
the chance you win $100
before you lose everything
594
00:37:17,250 --> 00:37:20,860
is at most 1 in 37,000.
595
00:37:20,860 --> 00:37:22,650
Now, we can plug in
some other values
596
00:37:22,650 --> 00:37:27,081
just for fun--
different values of m.
597
00:37:32,380 --> 00:37:36,350
If you thought 1 in
37,000 was unlikely,
598
00:37:36,350 --> 00:37:44,380
the chance of winning
$1,000, or 1,000 bets worth
599
00:37:44,380 --> 00:37:50,870
before you're broke-- that's
less than 9/10 to the 1,000.
600
00:37:50,870 --> 00:37:57,090
That's less than 2 times
10 to the minus 46--
601
00:37:57,090 --> 00:37:59,760
really, really, really unlikely.
602
00:37:59,760 --> 00:38:06,400
Even winning $10 is not likely.
603
00:38:06,400 --> 00:38:07,650
Just plug in the numbers.
604
00:38:07,650 --> 00:38:13,790
The probability you win
$10 betting $1 at a time
605
00:38:13,790 --> 00:38:18,470
is less than 9/10
to the 10th power.
606
00:38:18,470 --> 00:38:24,150
That's less than 0.35.
607
00:38:24,150 --> 00:38:30,710
You can come to the casino with
$10 million, bet $1 at a time,
608
00:38:30,710 --> 00:38:34,390
and you quit if you just
get up 10 bets-- get up $10.
609
00:38:34,390 --> 00:38:38,740
The chance you get up $10 before
you lose $10 million is about 1
610
00:38:38,740 --> 00:38:42,100
in 3 you're twice as
likely to lose $10 million
611
00:38:42,100 --> 00:38:44,400
as you are to win 10.
612
00:38:44,400 --> 00:38:46,780
That just seems weird, right?
613
00:38:46,780 --> 00:38:49,250
Because it's almost a fair game.
614
00:38:49,250 --> 00:38:51,390
It's almost 50-50.
615
00:38:51,390 --> 00:38:52,820
Any questions
about the analysis?
616
00:38:55,860 --> 00:38:59,710
Yes, I find that shocking.
617
00:38:59,710 --> 00:39:02,580
Just the intuition would
seem say otherwise.
618
00:39:02,580 --> 00:39:03,990
So I guess there's a moral here.
619
00:39:03,990 --> 00:39:05,880
If you're going to
gamble, learn how
620
00:39:05,880 --> 00:39:09,050
to count cards in blackjack,
or some game where
621
00:39:09,050 --> 00:39:10,722
you can make it even.
622
00:39:10,722 --> 00:39:12,680
Because even in a game
where it's pretty close,
623
00:39:12,680 --> 00:39:14,580
you're doomed.
624
00:39:14,580 --> 00:39:18,710
You're just never
going to go home happy.
625
00:39:18,710 --> 00:39:21,900
Now, if you could
have a fair game,
626
00:39:21,900 --> 00:39:24,780
the world changes-- much
better circumstance.
627
00:39:24,780 --> 00:39:28,800
So actually, let's do the
same analysis for a fair game,
628
00:39:28,800 --> 00:39:31,680
because that's where our
intuition really comes from.
629
00:39:31,680 --> 00:39:34,550
Because we're thinking of
this game as almost fair.
630
00:39:34,550 --> 00:39:38,700
And in a fair game, the answer's
going to be very different.
631
00:39:38,700 --> 00:39:41,320
And it all goes back to the
recurrence and the roots
632
00:39:41,320 --> 00:39:43,490
of the characteristic equation.
633
00:39:43,490 --> 00:39:49,380
Because in a fair
game, p is 1/2.
634
00:39:53,490 --> 00:39:56,290
And then you have a double root.
635
00:39:56,290 --> 00:40:04,970
1 minus 1/2 over 1/2 equals 1,
and that means a double root
636
00:40:04,970 --> 00:40:07,570
at 1.
637
00:40:07,570 --> 00:40:10,390
And that changes everything.
638
00:40:10,390 --> 00:40:16,120
So let's go through now
and do all this analysis
639
00:40:16,120 --> 00:40:17,440
in the case of a fair game.
640
00:40:20,140 --> 00:40:22,180
And this will give us
practice with double roots
641
00:40:22,180 --> 00:40:23,630
and recurrences.
642
00:40:23,630 --> 00:40:25,939
Because as you see
now, it does happen.
643
00:40:30,930 --> 00:40:33,866
Let's figure out the chance
that we go home a winner.
644
00:40:38,490 --> 00:40:40,360
OK, so let's see.
645
00:40:40,360 --> 00:40:44,766
In this case, we know the roots.
646
00:40:44,766 --> 00:40:46,580
Can anybody tell me
what formula we're
647
00:40:46,580 --> 00:40:49,870
going to use for the solution?
648
00:40:49,870 --> 00:40:52,230
Got a double root at 1.
649
00:40:52,230 --> 00:40:54,170
So there's going to
be a 1 to the n here.
650
00:40:56,750 --> 00:40:58,590
I don't just put a
constant A in front.
651
00:40:58,590 --> 00:41:00,744
What do I do with a double root?
652
00:41:00,744 --> 00:41:01,660
AUDIENCE: [INAUDIBLE].
653
00:41:01,660 --> 00:41:02,260
AUDIENCE: A n.
654
00:41:02,260 --> 00:41:03,430
PROFESSOR: What is it?
655
00:41:03,430 --> 00:41:04,230
AUDIENCE: A n.
656
00:41:04,230 --> 00:41:06,560
PROFESSOR: A n-- not quite A n.
657
00:41:06,560 --> 00:41:08,200
You got an A n here.
658
00:41:08,200 --> 00:41:09,080
AUDIENCE: Plus B.
659
00:41:09,080 --> 00:41:13,280
PROFESSOR: Plus B-- that's
what you do for a double root,
660
00:41:13,280 --> 00:41:17,340
because you make a first
degree polynomial in n here.
661
00:41:17,340 --> 00:41:18,782
So we plug that in.
662
00:41:18,782 --> 00:41:20,240
The root's at 1,
so it's real easy.
663
00:41:20,240 --> 00:41:21,910
The solution's really easy now.
664
00:41:21,910 --> 00:41:23,340
No messy powers or anything.
665
00:41:23,340 --> 00:41:28,487
It's just A n plus B. And
I can figure out A and B
666
00:41:28,487 --> 00:41:29,695
from the boundary conditions.
667
00:41:36,550 --> 00:41:40,590
All right, X0 is 0.
668
00:41:40,590 --> 00:41:46,280
X 0 is just B, because
it's A times 0 goes away.
669
00:41:46,280 --> 00:41:48,120
And that means that B equals 0.
670
00:41:48,120 --> 00:41:50,960
This is getting really simple.
671
00:41:50,960 --> 00:41:56,850
1 is X T. And that's
A plus B, but B was 0.
672
00:41:56,850 --> 00:42:03,620
So that's A times 1
plus B. That's just A.
673
00:42:03,620 --> 00:42:11,600
It means A equals A n.
674
00:42:11,600 --> 00:42:13,270
Good, n's not 1.
675
00:42:13,270 --> 00:42:15,410
N's T. So it's A T plus B.
676
00:42:15,410 --> 00:42:17,550
This is A T here.
677
00:42:17,550 --> 00:42:19,280
So A T equals 1.
678
00:42:19,280 --> 00:42:23,780
That means A is 1 over T.
679
00:42:23,780 --> 00:42:32,854
All right, that means that
X n is n over T. And T
680
00:42:32,854 --> 00:42:33,820
is the total.
681
00:42:33,820 --> 00:42:36,140
The top limit is n plus
m, because you quit
682
00:42:36,140 --> 00:42:38,090
if you get ahead m dollars.
683
00:42:38,090 --> 00:42:44,030
This is just now
n over n plus m.
684
00:42:44,030 --> 00:42:47,880
All right, so let's
write that down.
685
00:42:47,880 --> 00:42:48,515
It's a theorem.
686
00:42:53,450 --> 00:42:58,920
If p is 1/2, i.e., you
have a fair game, then
687
00:42:58,920 --> 00:43:04,580
the probability
you win m dollars
688
00:43:04,580 --> 00:43:15,270
before you lose n dollars
is just n over n plus m.
689
00:43:15,270 --> 00:43:17,590
And this might fit
the intuition better.
690
00:43:20,700 --> 00:43:28,810
So for the mother-in-law
strategy, if m is 100,
691
00:43:28,810 --> 00:43:34,580
and n is 1,000, what's
the probability you
692
00:43:34,580 --> 00:43:36,460
win-- you go home a winner?
693
00:43:42,230 --> 00:43:48,904
Yeah, 1,000 over 1,000 plus 100.
694
00:43:48,904 --> 00:43:54,490
1,000 over 1,000 is 10 over 11.
695
00:43:54,490 --> 00:43:57,850
So she does go home happy most
of the time-- 10 out of 11
696
00:43:57,850 --> 00:44:00,410
nights-- if she's
playing a fair game.
697
00:44:02,970 --> 00:44:06,920
Any questions about that?
698
00:44:06,920 --> 00:44:13,630
So the trouble we get into
here is that the fair game
699
00:44:13,630 --> 00:44:16,890
results match our intuition.
700
00:44:16,890 --> 00:44:20,520
You know if you have 10 times
as much money in a fair game,
701
00:44:20,520 --> 00:44:23,660
you'd expect to go home
happy 10 out of 11 nights.
702
00:44:23,660 --> 00:44:24,790
That makes a lot of sense.
703
00:44:24,790 --> 00:44:28,660
You go home happy 10, and
then you lose the 11th.
704
00:44:28,660 --> 00:44:30,790
That's a 10 to 1 ratio,
which is the money
705
00:44:30,790 --> 00:44:33,001
you brought into the game.
706
00:44:33,001 --> 00:44:36,160
The trouble we get
into is, the fair game
707
00:44:36,160 --> 00:44:38,790
is very close to the real game.
708
00:44:38,790 --> 00:44:42,120
Instead of 50-50, it's 47-53.
709
00:44:42,120 --> 00:44:44,310
And so our intuition
says the results--
710
00:44:44,310 --> 00:44:46,780
the probability of going
home happy in a fair game--
711
00:44:46,780 --> 00:44:49,200
should be close to the
probability of going
712
00:44:49,200 --> 00:44:51,180
home happy in the real game.
713
00:44:51,180 --> 00:44:52,340
And that's not true.
714
00:44:52,340 --> 00:44:55,990
There's a discontinuity here
because of the double root.
715
00:44:55,990 --> 00:44:58,350
And the character
completely changes.
716
00:44:58,350 --> 00:45:01,380
So instead of being
close to 10 out of 11,
717
00:45:01,380 --> 00:45:04,140
you're down there
at 1 in 37,000--
718
00:45:04,140 --> 00:45:07,640
completely different behavior.
719
00:45:07,640 --> 00:45:12,580
OK, any questions?
720
00:45:12,580 --> 00:45:14,398
All right, so let me
give you an-- yeah.
721
00:45:14,398 --> 00:45:17,540
AUDIENCE: So what
happens if you make n 1,
722
00:45:17,540 --> 00:45:20,520
and then you do that repeatedly?
723
00:45:20,520 --> 00:45:23,280
PROFESSOR: Now, if
I did n equals 1,
724
00:45:23,280 --> 00:45:24,750
I could use that
as an upper bound,
725
00:45:24,750 --> 00:45:28,080
and it's not so
interesting as, say, 90%.
726
00:45:28,080 --> 00:45:30,890
But I would actually go
plug it back in here.
727
00:45:30,890 --> 00:45:33,300
So this would be n plus
1, and it would depend
728
00:45:33,300 --> 00:45:34,890
how much money I brought.
729
00:45:34,890 --> 00:45:39,640
But there is a pretty good
chance I go home a winner for m
730
00:45:39,640 --> 00:45:41,230
equals 1.
731
00:45:41,230 --> 00:45:43,940
Because I've got a pretty
good chance that I either--
732
00:45:43,940 --> 00:45:46,090
47% chance I win the first time.
733
00:45:46,090 --> 00:45:47,752
Then I go home happy.
734
00:45:47,752 --> 00:45:50,790
If I lost the first time, now
I've just got to win twice.
735
00:45:50,790 --> 00:45:52,720
And I might win twice in a row.
736
00:45:52,720 --> 00:45:55,330
That'll happen about
20% of the time.
737
00:45:55,330 --> 00:45:58,940
If I lose that, now I've
got to win three in a row.
738
00:45:58,940 --> 00:46:02,150
That'll happen around
10% of the time.
739
00:46:02,150 --> 00:46:05,540
So I've got 10 plus
20 plus almost 50.
740
00:46:05,540 --> 00:46:08,010
Most of the time, I'm going
to go home happy if I just
741
00:46:08,010 --> 00:46:10,460
have to get ahead by $1.
742
00:46:10,460 --> 00:46:12,460
But it doesn't take
much more than one
743
00:46:12,460 --> 00:46:14,660
before you're not
likely to go home happy.
744
00:46:14,660 --> 00:46:19,050
Getting ahead 10 is not
going to happen, very likely.
745
00:46:19,050 --> 00:46:21,930
Now, you want to
recurse on that?
746
00:46:21,930 --> 00:46:25,490
I'm pretty likely
to get ahead by one.
747
00:46:25,490 --> 00:46:26,990
Well, OK, get ahead by one.
748
00:46:26,990 --> 00:46:29,639
I'm pretty likely
to do it again.
749
00:46:29,639 --> 00:46:30,430
And I did it again.
750
00:46:30,430 --> 00:46:32,320
Now I'm pretty likely
to do it again.
751
00:46:32,320 --> 00:46:33,945
And there's this
thing called induction
752
00:46:33,945 --> 00:46:35,270
that we worried a lot about.
753
00:46:35,270 --> 00:46:38,130
So by induction, are we likely
to go home happy with 10?
754
00:46:38,130 --> 00:46:43,460
No, because every time you
don't get there, you're dead.
755
00:46:43,460 --> 00:46:45,720
You had a little chance of
dying and not reaching one,
756
00:46:45,720 --> 00:46:48,251
and a little chance of dying
and not going from one to two.
757
00:46:48,251 --> 00:46:50,000
And you add up all
those chances of dying,
758
00:46:50,000 --> 00:46:52,440
and you're toast, because
that'll be adding up
759
00:46:52,440 --> 00:46:56,317
to everything, pretty much.
760
00:46:56,317 --> 00:46:57,400
So that's a good question.
761
00:46:57,400 --> 00:46:59,410
If you're likely
to get up by one,
762
00:46:59,410 --> 00:47:01,839
why aren't you likely
to get up by 10?
763
00:47:01,839 --> 00:47:02,880
It doesn't work that way.
764
00:47:02,880 --> 00:47:05,300
That's a great question.
765
00:47:05,300 --> 00:47:12,130
Let me show you the phenomenon
that's going on here, as
766
00:47:12,130 --> 00:47:15,300
to why it works out this way.
767
00:47:15,300 --> 00:47:16,644
We had the math.
768
00:47:16,644 --> 00:47:17,810
So we looked at it that way.
769
00:47:17,810 --> 00:47:20,810
We notice that one case is a
double root and the other case
770
00:47:20,810 --> 00:47:22,200
isn't.
771
00:47:22,200 --> 00:47:23,640
And that exponential,
in the case
772
00:47:23,640 --> 00:47:25,670
where you didn't have
that second root at 1
773
00:47:25,670 --> 00:47:28,590
makes an enormous difference.
774
00:47:28,590 --> 00:47:32,800
Qualitatively, we can
draw the two cases.
775
00:47:32,800 --> 00:47:43,010
So in the case of an
unbiased or fair game,
776
00:47:43,010 --> 00:47:47,640
if we track what's
going on over time,
777
00:47:47,640 --> 00:47:54,265
and we start with n dollars,
sort of this is our baseline.
778
00:47:57,350 --> 00:48:02,190
And here's our
target-- T is n plus m.
779
00:48:02,190 --> 00:48:04,230
And so we quit if
we ever get here.
780
00:48:04,230 --> 00:48:06,530
And we quit if we
ever hit the bottom.
781
00:48:06,530 --> 00:48:08,920
And we've got a random walk.
782
00:48:08,920 --> 00:48:15,200
It's going around, just
doing this kind of stuff.
783
00:48:15,200 --> 00:48:19,150
And eventually, it's going to
hit one of these boundaries.
784
00:48:19,150 --> 00:48:23,580
And if m is small
compared to n, we're
785
00:48:23,580 --> 00:48:26,170
more likely to
hit this boundary.
786
00:48:26,170 --> 00:48:29,150
And in fact, the chance
we hit this boundary first
787
00:48:29,150 --> 00:48:31,120
is the ratio of these sizes.
788
00:48:31,120 --> 00:48:34,230
It's n over the total.
789
00:48:34,230 --> 00:48:38,300
It's the chance that
we hit that one first.
790
00:48:38,300 --> 00:48:42,300
Now in the biased case, the
picture looks different.
791
00:48:55,060 --> 00:49:02,342
So in the biased case--
so this is now biased.
792
00:49:02,342 --> 00:49:04,300
And we're going to assume
it's downward biased.
793
00:49:04,300 --> 00:49:05,425
You're more likely to lose.
794
00:49:09,960 --> 00:49:13,770
So you start at n, you've
got your boundary up here
795
00:49:13,770 --> 00:49:17,740
at T equals n plus m.
796
00:49:17,740 --> 00:49:21,190
Time is going this way.
797
00:49:21,190 --> 00:49:28,810
The problem is, you've got
a downward sort of baseline,
798
00:49:28,810 --> 00:49:33,080
because you expect to lose
a little bit each time.
799
00:49:33,080 --> 00:49:35,785
And so you're taking
this random walk.
800
00:49:38,360 --> 00:49:41,970
And you collide here.
801
00:49:41,970 --> 00:49:47,050
And these things are
known as the swings.
802
00:49:47,050 --> 00:49:48,795
This is known as the drift.
803
00:49:52,390 --> 00:49:55,930
And the drift downward
is 1 minus 2p.
804
00:49:55,930 --> 00:49:58,510
That's what you expect to
lose if you get the expected
805
00:49:58,510 --> 00:50:01,690
loss on each bet-- 1 minus 2p.
806
00:50:01,690 --> 00:50:04,800
Because you're going
to not be a fair game.
807
00:50:04,800 --> 00:50:06,810
This one has zero
drift up there.
808
00:50:06,810 --> 00:50:08,590
It stays steady.
809
00:50:08,590 --> 00:50:17,195
And in random walks, drift
outweighs the swings.
810
00:50:19,900 --> 00:50:21,420
These are the swings here.
811
00:50:21,420 --> 00:50:23,310
And they're random.
812
00:50:23,310 --> 00:50:24,980
The drift is deterministic.
813
00:50:24,980 --> 00:50:26,950
It's steadily going down.
814
00:50:26,950 --> 00:50:28,890
And so almost always
in a random walk,
815
00:50:28,890 --> 00:50:32,520
the drift totally
takes over the swings.
816
00:50:32,520 --> 00:50:34,400
The swings are small
compared to what you're
817
00:50:34,400 --> 00:50:37,730
losing on a steady basis.
818
00:50:37,730 --> 00:50:41,260
And that's why you're so
much more likely to lose when
819
00:50:41,260 --> 00:50:44,110
you have the drift downward.
820
00:50:46,670 --> 00:50:49,450
Just as an example, maybe
putting some numbers
821
00:50:49,450 --> 00:50:49,950
around that.
822
00:50:53,680 --> 00:50:55,700
The swings are the
same in both cases,
823
00:50:55,700 --> 00:50:59,180
So that gives you some
qualification for how big
824
00:50:59,180 --> 00:51:00,308
the swings tend to be.
825
00:51:03,660 --> 00:51:06,365
We can sort of do that with
standard deviation notation.
826
00:51:09,380 --> 00:51:17,510
After X bets or X steps,
the amount you've drifted,
827
00:51:17,510 --> 00:51:21,530
or the expected
losses, 1 minus 2p
828
00:51:21,530 --> 00:51:25,940
X. Maybe we should
just understand
829
00:51:25,940 --> 00:51:27,300
why this is the case.
830
00:51:27,300 --> 00:51:37,960
The expected return on a
bet is 1 with probability p,
831
00:51:37,960 --> 00:51:41,650
and minus 1 with
probability 1 minus p.
832
00:51:41,650 --> 00:51:46,046
And so that is-- did
I get that right?
833
00:51:46,046 --> 00:51:49,730
I think that's right.
834
00:51:49,730 --> 00:51:52,200
Oh, expected loss--
[INAUDIBLE] drifts down.
835
00:51:52,200 --> 00:51:54,569
Instead of expected
return, let's do the loss,
836
00:51:54,569 --> 00:51:55,610
because that's the drift.
837
00:51:55,610 --> 00:51:57,270
It's a downward thing.
838
00:51:57,270 --> 00:52:01,910
So the expected loss-- now
you lose $1 with 1 minus p.
839
00:52:04,610 --> 00:52:07,670
And you gain $1, which
is negative loss,
840
00:52:07,670 --> 00:52:10,050
with probability p.
841
00:52:10,050 --> 00:52:14,570
And so you get 1 minus
p minus p is 1 minus 2p.
842
00:52:14,570 --> 00:52:16,370
So that's your expected loss.
843
00:52:16,370 --> 00:52:21,300
Your expected winnings
are the negative of that.
844
00:52:21,300 --> 00:52:25,700
So after x steps, you
expect to lose-- well,
845
00:52:25,700 --> 00:52:27,700
I just add up the
linearity of expectation.
846
00:52:27,700 --> 00:52:30,950
You expect to lose
this much x times.
847
00:52:30,950 --> 00:52:32,480
So that's your expected drift.
848
00:52:32,480 --> 00:52:36,440
You're expected
to lose that much.
849
00:52:36,440 --> 00:52:38,950
Now, the swing--
and we won't prove
850
00:52:38,950 --> 00:52:44,670
this-- the swing is expected
to be square root of x
851
00:52:44,670 --> 00:52:45,480
times a constant.
852
00:52:45,480 --> 00:52:47,670
So I've used the
theta notation here.
853
00:52:47,670 --> 00:52:50,230
And the constant is small.
854
00:52:50,230 --> 00:52:54,090
If I take x consecutive
bets for $1,
855
00:52:54,090 --> 00:52:57,610
I'm very likely to be
about square root of x
856
00:52:57,610 --> 00:53:01,510
off of the expected drift.
857
00:53:01,510 --> 00:53:05,040
And you can see that
this is square root.
858
00:53:05,040 --> 00:53:06,550
That is linear.
859
00:53:06,550 --> 00:53:10,250
So this totally dominates that.
860
00:53:10,250 --> 00:53:14,110
So the swings are generally
not enough to save you.
861
00:53:14,110 --> 00:53:17,300
And so you're just going to
cruise downward and crash,
862
00:53:17,300 --> 00:53:18,136
almost surely.
863
00:53:21,330 --> 00:53:25,160
OK, any questions about that?
864
00:53:31,360 --> 00:53:34,430
All right, so we figured out
the probability of winning m
865
00:53:34,430 --> 00:53:36,420
dollars before going broke.
866
00:53:36,420 --> 00:53:40,230
That's done with.
867
00:53:40,230 --> 00:53:43,060
Now, this means it's
logical to conclude
868
00:53:43,060 --> 00:53:47,310
you're likely go home
broke in an unfair game.
869
00:53:47,310 --> 00:53:50,820
Actually, before we do
that, there's one other case
870
00:53:50,820 --> 00:53:52,760
we've got to rule out.
871
00:53:52,760 --> 00:53:56,512
We've proved you're likely
not to go home a winner.
872
00:53:56,512 --> 00:53:58,720
Does that necessarily mean
you're likely to go broke?
873
00:53:58,720 --> 00:54:00,845
I've been saying that, but
there's some other thing
874
00:54:00,845 --> 00:54:03,240
we should check.
875
00:54:03,240 --> 00:54:07,484
What's one way you
might not go home broke?
876
00:54:07,484 --> 00:54:08,400
AUDIENCE: [INAUDIBLE].
877
00:54:08,400 --> 00:54:09,385
PROFESSOR: What is it?
878
00:54:09,385 --> 00:54:09,780
AUDIENCE: You don't go home.
879
00:54:09,780 --> 00:54:11,220
PROFESSOR: You don't go home.
880
00:54:11,220 --> 00:54:13,520
And why would you not go home?
881
00:54:13,520 --> 00:54:14,020
Yeah?
882
00:54:14,020 --> 00:54:15,395
AUDIENCE: You're
playing forever.
883
00:54:15,395 --> 00:54:16,990
PROFESSOR: You're
playing forever-- we
884
00:54:16,990 --> 00:54:19,840
didn't rule out that case--
you're playing forever.
885
00:54:19,840 --> 00:54:22,790
But it turns out, if you
did the same analysis,
886
00:54:22,790 --> 00:54:25,200
you can analyze the probability
of going home broke.
887
00:54:25,200 --> 00:54:27,366
And when you add it to the
probability of going home
888
00:54:27,366 --> 00:54:30,320
a winner, it adds to 1,
which means the probability
889
00:54:30,320 --> 00:54:33,480
playing forever is 0.
890
00:54:33,480 --> 00:54:35,757
Now, there are sample points
where you play forever.
891
00:54:35,757 --> 00:54:37,590
But when you add up all
those sample points,
892
00:54:37,590 --> 00:54:41,210
if their probability
is 0, we ignore them.
893
00:54:41,210 --> 00:54:42,950
And we say it can't happen.
894
00:54:42,950 --> 00:54:46,379
Now, we're bordering
on philosophy here,
895
00:54:46,379 --> 00:54:47,920
because there is a
sample point here.
896
00:54:47,920 --> 00:54:50,640
You could win, lose, win,
lose, win, lose forever.
897
00:54:50,640 --> 00:54:54,305
But because you add them
all up at 0, measure theory
898
00:54:54,305 --> 00:54:56,430
and some math we're not
going to get into tells you
899
00:54:56,430 --> 00:54:58,010
it doesn't happen.
900
00:54:58,010 --> 00:55:00,430
It's probability 1 you're
a winner or a loser.
901
00:55:07,430 --> 00:55:09,760
All right, so I'm
not going to prove
902
00:55:09,760 --> 00:55:13,350
that the probability
you play forever is 0.
903
00:55:24,250 --> 00:55:27,660
But let's look at
how long you play.
904
00:55:27,660 --> 00:55:32,230
How long does it take you to
go home one way or another-- go
905
00:55:32,230 --> 00:55:33,750
broke?
906
00:55:33,750 --> 00:55:36,440
And to do this, we're going
to set up another recurrence.
907
00:55:39,440 --> 00:55:42,160
So we know eventually
we hit a boundary.
908
00:55:42,160 --> 00:55:44,160
I want to know how
many bets does it
909
00:55:44,160 --> 00:55:48,100
take to hit the boundary?
910
00:55:48,100 --> 00:55:50,530
How long do we get to play
before we go home unhappy?
911
00:55:53,270 --> 00:55:59,150
So S will be the number of
steps until we hit a boundary.
912
00:56:04,320 --> 00:56:06,820
And I want to know the
expected number-- I'll call it
913
00:56:06,820 --> 00:56:10,464
E sub n here-- is the
expected value of S given
914
00:56:10,464 --> 00:56:11,630
that I start with n dollars.
915
00:56:14,884 --> 00:56:16,300
I mean, the reason
you could think
916
00:56:16,300 --> 00:56:18,550
about this is, we know
we're going to go home
917
00:56:18,550 --> 00:56:20,410
broke-- pretty likely.
918
00:56:20,410 --> 00:56:22,810
Do we at least have some
fun in the meantime?
919
00:56:22,810 --> 00:56:25,640
Do we get a lot gambling in
and free drinks, or whatever,
920
00:56:25,640 --> 00:56:28,700
before we're killed here?
921
00:56:31,590 --> 00:56:33,210
Now, this also has a recurrence.
922
00:56:33,210 --> 00:56:35,030
And I'm going to show
you what it is, then
923
00:56:35,030 --> 00:56:37,940
prove that that's correct.
924
00:56:37,940 --> 00:56:42,610
So I claim that
the expected number
925
00:56:42,610 --> 00:56:44,600
of steps given we
start with n dollars
926
00:56:44,600 --> 00:56:51,500
is 0 if we start with no money,
because we are already broke.
927
00:56:51,500 --> 00:56:57,370
It's 0 if we start
with T dollars,
928
00:56:57,370 --> 00:56:59,110
because then we
just go home happy.
929
00:56:59,110 --> 00:57:03,290
There's no bets, because we've
already hit the upper boundary.
930
00:57:03,290 --> 00:57:09,490
And the interesting case will be
it's 1 plus p times E n minus 1
931
00:57:09,490 --> 00:57:17,980
plus 1 minus p-- oops, n plus
1-- 1 minus p E n minus 1,
932
00:57:17,980 --> 00:57:21,423
if we start with
between 0 and T dollars.
933
00:57:26,980 --> 00:57:30,650
OK, so let's prove that.
934
00:57:33,400 --> 00:57:38,660
Actually, the proof is exactly
the same as the last one.
935
00:57:38,660 --> 00:57:40,194
So I don't think
I need to do it.
936
00:57:44,300 --> 00:57:48,370
The proof is pretty simple,
because we look at two cases.
937
00:57:48,370 --> 00:57:52,150
You win the first bet--
happens with probability p.
938
00:57:52,150 --> 00:57:55,250
And then you're starting
with n plus $1 over again.
939
00:57:55,250 --> 00:57:58,050
Or you lose the
first bet-- happens
940
00:57:58,050 --> 00:58:00,360
with probability 1 minus p.
941
00:58:00,360 --> 00:58:03,890
And you're starting
over with n minus $1
942
00:58:03,890 --> 00:58:05,705
now-- same as last time.
943
00:58:05,705 --> 00:58:07,080
In fact, this
whole recurrence is
944
00:58:07,080 --> 00:58:11,690
identical to last time
except for one thing.
945
00:58:11,690 --> 00:58:13,914
What's the one thing
that's different now?
946
00:58:13,914 --> 00:58:14,870
AUDIENCE: [INAUDIBLE].
947
00:58:14,870 --> 00:58:15,994
PROFESSOR: What is it?
948
00:58:15,994 --> 00:58:18,780
AUDIENCE: You have [INAUDIBLE].
949
00:58:18,780 --> 00:58:20,738
PROFESSOR: You have--
950
00:58:20,738 --> 00:58:23,208
AUDIENCE: So it's not
[INAUDIBLE] any more.
951
00:58:23,208 --> 00:58:24,690
PROFESSOR: That's different.
952
00:58:24,690 --> 00:58:26,920
There's another difference.
953
00:58:26,920 --> 00:58:30,100
That's one difference that's
going to make it inhomogeneous.
954
00:58:30,100 --> 00:58:31,409
That's sort of a pain.
955
00:58:31,409 --> 00:58:33,200
What's the other
difference from last time?
956
00:58:33,200 --> 00:58:35,025
This part's the same otherwise.
957
00:58:35,025 --> 00:58:35,900
AUDIENCE: Boundaries.
958
00:58:35,900 --> 00:58:36,400
PROFESSOR: What is it?
959
00:58:36,400 --> 00:58:37,650
AUDIENCE: Boundary conditions.
960
00:58:37,650 --> 00:58:40,120
PROFESSOR: Boundary conditions--
that was a 1 before.
961
00:58:40,120 --> 00:58:42,290
Now it's a 0.
962
00:58:42,290 --> 00:58:46,420
OK, so a little change
here, and I added a 1 here.
963
00:58:46,420 --> 00:58:50,041
But that's going to make it
a pretty different answer.
964
00:58:50,041 --> 00:58:51,540
So let's see what
the recurrence is.
965
00:58:51,540 --> 00:58:57,310
I'll rearrange terms here
to put it into recurrence.
966
00:58:57,310 --> 00:59:07,650
I get p E sub n plus 1 minus
E n plus 1 minus p E n minus 1
967
00:59:07,650 --> 00:59:10,710
equals minus 1, not 0.
968
00:59:10,710 --> 00:59:15,570
And the boundary conditions
are E 0 is 0 and E T is 0.
969
00:59:18,210 --> 00:59:21,550
OK, what's the
first thing you do
970
00:59:21,550 --> 00:59:26,020
when you have an inhomogeneous
linear recurrence?
971
00:59:26,020 --> 00:59:28,920
Solve the homogeneous one.
972
00:59:28,920 --> 00:59:32,050
And the answer there-- well,
it's the same as before.
973
00:59:32,050 --> 00:59:33,990
This is the part we analyzed.
974
00:59:33,990 --> 00:59:36,650
And we'll do it for
the case when p is not
975
00:59:36,650 --> 00:59:39,770
1/2-- so the unfair game.
976
00:59:39,770 --> 00:59:47,270
So the homogeneous
solution is E n
977
00:59:47,270 --> 00:59:51,870
just from before-- same
thing-- 1 minus p over p
978
00:59:51,870 --> 00:59:55,720
to the n plus B. And this
is the case with two roots.
979
00:59:55,720 --> 00:59:56,842
p does not equal 1/2.
980
01:00:00,620 --> 01:00:07,634
What's the next thing you do
for inhomogeneous recurrence?
981
01:00:07,634 --> 01:00:11,620
Are we plugging in
boundary conditions yet?
982
01:00:11,620 --> 01:00:12,120
No.
983
01:00:12,120 --> 01:00:14,840
So what do I do next?
984
01:00:14,840 --> 01:00:15,860
Particular solution.
985
01:00:21,160 --> 01:00:25,660
And what's my first guess?
986
01:00:25,660 --> 01:00:31,780
We have the recurrence
like this here.
987
01:00:34,590 --> 01:00:37,960
What do I guess for E n?
988
01:00:37,960 --> 01:00:42,610
I'm trying to guess something
that looks like that.
989
01:00:42,610 --> 01:00:45,380
So what do I guess?
990
01:00:45,380 --> 01:00:47,740
Constant, yeah.
991
01:00:47,740 --> 01:00:48,420
That's a scalar.
992
01:00:48,420 --> 01:00:51,120
I just guess a constant.
993
01:00:51,120 --> 01:00:57,420
And if I plug a constant a
into here, it's going to fail.
994
01:00:57,420 --> 01:01:01,730
Because I'll just
pull the a out.
995
01:01:01,730 --> 01:01:05,740
I'll get p minus 1
plus 1 minus p is 0,
996
01:01:05,740 --> 01:01:07,640
and 0 doesn't equal minus 1.
997
01:01:07,640 --> 01:01:08,220
So it fails.
998
01:01:12,240 --> 01:01:13,290
So I guess again.
999
01:01:13,290 --> 01:01:16,380
What do I guess next time?
1000
01:01:16,380 --> 01:01:19,990
a n plus b.
1001
01:01:19,990 --> 01:01:22,580
All right, and I
don't think I'll
1002
01:01:22,580 --> 01:01:28,430
drag you through all the
algebra for that, but it works.
1003
01:01:28,430 --> 01:01:35,310
And when you do it, you
find that a is minus 1
1004
01:01:35,310 --> 01:01:38,100
over 2p minus 1.
1005
01:01:38,100 --> 01:01:39,470
And b could be anything.
1006
01:01:39,470 --> 01:01:44,080
So let me just rewrite
this as 1 over 1 minus 2p.
1007
01:01:44,080 --> 01:01:46,170
And b can be anything, so
we'll set b equal to 0.
1008
01:01:49,490 --> 01:01:52,910
So we've got our
particular solution.
1009
01:01:52,910 --> 01:01:54,520
It's not hard to
go compute that.
1010
01:01:54,520 --> 01:01:57,402
You just plug it
back in and solve.
1011
01:02:04,642 --> 01:02:06,850
Now we add them together to
get the general solution.
1012
01:02:18,830 --> 01:02:25,440
This is A n plus B. B was 0, and
here's A as 1 over 1 minus 2p.
1013
01:02:25,440 --> 01:02:29,270
And now what do we do to finish?
1014
01:02:29,270 --> 01:02:33,700
I've got my general
solution here
1015
01:02:33,700 --> 01:02:37,537
by adding up the homogeneous
and the particular solution.
1016
01:02:37,537 --> 01:02:38,870
Plug in the boundary conditions.
1017
01:02:45,680 --> 01:02:49,789
All right, I'm not going to drag
you through solving this case,
1018
01:02:49,789 --> 01:02:51,330
but I'm going to
show you the answer.
1019
01:02:55,430 --> 01:03:03,480
E n equals n over 1 minus 2p
minus T, the upper boundary,
1020
01:03:03,480 --> 01:03:13,400
over 1 minus 2p times 1 minus p
over p to the n minus 1 over 1
1021
01:03:13,400 --> 01:03:17,930
minus p over p to the T minus 1.
1022
01:03:17,930 --> 01:03:20,670
So actually, this looks a little
familiar from the last time
1023
01:03:20,670 --> 01:03:23,742
when we did this recurrence,
figuring out the probability we
1024
01:03:23,742 --> 01:03:24,450
go home a winner.
1025
01:03:24,450 --> 01:03:27,630
Here this is the
expected number of steps
1026
01:03:27,630 --> 01:03:30,900
to hit a boundary, to go home.
1027
01:03:30,900 --> 01:03:33,130
If we plug in the values,
it's a little hairy,
1028
01:03:33,130 --> 01:03:35,850
but you can compute it.
1029
01:03:35,850 --> 01:03:42,860
So for example, if m
is 100, n is 1,000,
1030
01:03:42,860 --> 01:03:47,160
T would be 1,100 in that case.
1031
01:03:47,160 --> 01:03:52,210
p is 9/19 playing roulette.
1032
01:03:52,210 --> 01:03:59,580
Then the expected number of
bets before you have to go home
1033
01:03:59,580 --> 01:04:13,480
is 1,900 from this part,
minus 0.56 from that part.
1034
01:04:13,480 --> 01:04:17,570
So actually 19,000, sorry.
1035
01:04:17,570 --> 01:04:24,580
So it's very close to 19,000
bets you've got to make.
1036
01:04:24,580 --> 01:04:28,230
So it takes a long
time to lose $1,000.
1037
01:04:28,230 --> 01:04:32,410
And it sort of comes
very close to the answer
1038
01:04:32,410 --> 01:04:34,860
you would have guessed
without thinking and solving
1039
01:04:34,860 --> 01:04:36,120
the recurrence.
1040
01:04:36,120 --> 01:04:41,350
If you expect to lose
1 minus 2p every bet,
1041
01:04:41,350 --> 01:04:44,160
and you want to know how long
the expected time to lose
1042
01:04:44,160 --> 01:04:46,890
n dollars, you might
well have said,
1043
01:04:46,890 --> 01:04:51,210
I think it's going to be n over
the amount I lose every time.
1044
01:04:51,210 --> 01:04:54,460
That would be
wrong, technically,
1045
01:04:54,460 --> 01:04:57,400
because you'd have left
off this nasty thing.
1046
01:04:57,400 --> 01:05:00,780
But this nasty thing doesn't
make much of a real difference,
1047
01:05:00,780 --> 01:05:04,245
because it goes to 0 really
fast for any numbers like 100
1048
01:05:04,245 --> 01:05:06,570
and 1,000-- makes no
difference at all.
1049
01:05:06,570 --> 01:05:08,380
So the intuition in
that case comes out
1050
01:05:08,380 --> 01:05:11,430
to be pretty close, even
though technically, it's
1051
01:05:11,430 --> 01:05:15,540
not exactly right.
1052
01:05:15,540 --> 01:05:21,460
Now, to see why this goes to
0, if T equals n plus m here--
1053
01:05:21,460 --> 01:05:25,350
this is n plus m--
and your upper limits,
1054
01:05:25,350 --> 01:05:31,040
say m goes to infinity--
it's 100 in this case-- then
1055
01:05:31,040 --> 01:05:34,770
that just zooms to 0, and
you're only left with that.
1056
01:05:34,770 --> 01:05:39,510
Which means that we can use
asymptotic notation here
1057
01:05:39,510 --> 01:05:42,782
to sort of characterize the
expected number of bets.
1058
01:05:48,020 --> 01:05:51,000
And it's totally
dominated by the drift.
1059
01:05:51,000 --> 01:05:58,120
So as m goes to infinity, the
expected time to live here
1060
01:05:58,120 --> 01:06:02,060
is tilde n over 1 minus 2p.
1061
01:06:02,060 --> 01:06:07,330
If you've got n dollars,
losing 1 minus 2p every time,
1062
01:06:07,330 --> 01:06:11,650
then you last for n
over 1 minus 2p steps.
1063
01:06:11,650 --> 01:06:17,180
OK, now, actually,
what situation in words
1064
01:06:17,180 --> 01:06:19,900
does m going to infinity mean?
1065
01:06:19,900 --> 01:06:23,200
Say I set m to be infinity?
1066
01:06:23,200 --> 01:06:26,810
What is that kind of
game if m is infinity?
1067
01:06:26,810 --> 01:06:28,120
How long am I playing now?
1068
01:06:28,120 --> 01:06:28,729
Yeah.
1069
01:06:28,729 --> 01:06:30,520
AUDIENCE: Now you're
playing for as long as
1070
01:06:30,520 --> 01:06:32,760
it takes you to lose
all of your money.
1071
01:06:32,760 --> 01:06:36,200
PROFESSOR: Yes, because there is
no stopping condition up here--
1072
01:06:36,200 --> 01:06:37,380
going home happy.
1073
01:06:37,380 --> 01:06:43,060
I'm going to play forever
or until I lose everything.
1074
01:06:43,060 --> 01:06:47,460
And this says how long
you expect to play.
1075
01:06:47,460 --> 01:06:51,220
It's a little less
than n over 1 minus 2p.
1076
01:06:51,220 --> 01:06:53,900
So if you play
until you go broke,
1077
01:06:53,900 --> 01:06:55,510
that's how long
you expect to play.
1078
01:06:59,950 --> 01:07:03,200
So that sort of makes
sense in that scenario.
1079
01:07:03,200 --> 01:07:06,570
That's not one where it
surprises you by intuition.
1080
01:07:06,570 --> 01:07:08,904
It is interesting to consider
the case of a fair game.
1081
01:07:08,904 --> 01:07:10,820
Because there's something
that's non-intuitive
1082
01:07:10,820 --> 01:07:12,570
that happens there.
1083
01:07:12,570 --> 01:07:14,290
So in a fair game, p is 1/2.
1084
01:07:17,350 --> 01:07:24,280
Now, if I plug in 1/2
here, well, I divide by 0.
1085
01:07:24,280 --> 01:07:27,730
I expect to play forever.
1086
01:07:27,730 --> 01:07:29,500
That's not a good way
to do the analysis,
1087
01:07:29,500 --> 01:07:30,950
that you get to a divide by 0.
1088
01:07:30,950 --> 01:07:33,580
Let's actually go
back and look at this
1089
01:07:33,580 --> 01:07:35,100
for the case when p is 1/2.
1090
01:07:37,769 --> 01:07:39,310
And see what happens
in a fair game--
1091
01:07:39,310 --> 01:07:43,250
how long you expect to
play in a fair game.
1092
01:07:43,250 --> 01:07:48,500
Then the homogeneous
solution is the simple case.
1093
01:07:48,500 --> 01:07:54,270
E is A n plus B. You
have a double root at 1,
1094
01:07:54,270 --> 01:07:57,340
which we don't have to
worry about 1 to the n.
1095
01:07:57,340 --> 01:08:03,550
When you do your
particular solution,
1096
01:08:03,550 --> 01:08:08,620
you'll try a single
scalar, and it fails.
1097
01:08:08,620 --> 01:08:11,730
I'll use lowercase a-- fails.
1098
01:08:11,730 --> 01:08:18,460
You will then try a degree one
polynomial, and that will fail.
1099
01:08:18,460 --> 01:08:21,010
What are you going to try next?
1100
01:08:21,010 --> 01:08:28,210
Second-degree polynomial,
and that will work.
1101
01:08:28,210 --> 01:08:34,100
OK, and the answer
you get when you
1102
01:08:34,100 --> 01:08:41,960
do that is that-- I'll
put the answer here.
1103
01:08:41,960 --> 01:08:47,450
It turns out that a is minus
1 and b and c can be 0.
1104
01:08:47,450 --> 01:08:49,470
So it's just going
to be minus n squared
1105
01:08:49,470 --> 01:08:51,660
for the particular solution.
1106
01:08:51,660 --> 01:08:59,931
That means your general solution
is A n plus B minus n squared.
1107
01:08:59,931 --> 01:09:01,389
Now you do your
boundary condition.
1108
01:09:06,660 --> 01:09:10,180
You have E 0 is 0.
1109
01:09:10,180 --> 01:09:11,359
Plug in 0 for n.
1110
01:09:11,359 --> 01:09:13,720
That's equal to B. So B is 0.
1111
01:09:13,720 --> 01:09:15,620
That's nice.
1112
01:09:15,620 --> 01:09:19,180
E T is 0.
1113
01:09:19,180 --> 01:09:25,029
And I plug in T here, I get
AT, B is 0 minus T squared.
1114
01:09:25,029 --> 01:09:28,160
So I solve for A here.
1115
01:09:28,160 --> 01:09:35,550
That means that A equals T. AT
squared minus T squared is 0.
1116
01:09:35,550 --> 01:09:38,439
A has to be T.
1117
01:09:38,439 --> 01:09:49,960
So that means that E n
is Tn minus n squared.
1118
01:09:49,960 --> 01:09:52,090
Now, T is the upper bound.
1119
01:09:52,090 --> 01:09:53,140
It's just n plus m.
1120
01:09:56,800 --> 01:10:00,880
n plus m times n
minus n squared--
1121
01:10:00,880 --> 01:10:01,925
this gets really simple.
1122
01:10:01,925 --> 01:10:03,190
The m squared cancels.
1123
01:10:03,190 --> 01:10:06,470
I just get n out.
1124
01:10:06,470 --> 01:10:08,820
That says if you're playing
a fair game, until you
1125
01:10:08,820 --> 01:10:14,400
win m or lose n, you expect
to play for nm steps, which
1126
01:10:14,400 --> 01:10:15,680
is really nice.
1127
01:10:15,680 --> 01:10:21,100
This is p is 1/2-- very clean.
1128
01:10:21,100 --> 01:10:25,940
Now, if you let m
equal to infinity,
1129
01:10:25,940 --> 01:10:29,680
you're going to expect
to play forever.
1130
01:10:29,680 --> 01:10:33,680
So with a fair game, if you
play until you're broke,
1131
01:10:33,680 --> 01:10:37,220
the expected number
of bets is infinite.
1132
01:10:37,220 --> 01:10:39,510
That's nice.
1133
01:10:39,510 --> 01:10:41,990
You can play forever
is the expectation.
1134
01:10:46,160 --> 01:10:50,430
Now, here's the weird thing.
1135
01:10:50,430 --> 01:10:53,420
If you expect to play
forever, does that
1136
01:10:53,420 --> 01:10:55,725
mean you're not likely
to go home broke?
1137
01:10:58,580 --> 01:11:02,410
You expect to play forever.
1138
01:11:02,410 --> 01:11:05,976
And as long as you're playing,
you're not going home broke.
1139
01:11:05,976 --> 01:11:07,850
Now, there's some chance
of going home broke,
1140
01:11:07,850 --> 01:11:10,330
because you might just lose
every bet-- not likely.
1141
01:11:15,930 --> 01:11:18,580
Here's the weird
thing-- the probability
1142
01:11:18,580 --> 01:11:24,460
you go home broke if you
play until you go broke is 1.
1143
01:11:24,460 --> 01:11:26,970
You will go home broke.
1144
01:11:26,970 --> 01:11:30,140
It's just that it
takes you expected
1145
01:11:30,140 --> 01:11:32,930
infinite amount
of time to do it--
1146
01:11:32,930 --> 01:11:36,350
sort of one of these weird
things in a fair game.
1147
01:11:36,350 --> 01:11:40,330
So here we proved the
expected number bets is nm.
1148
01:11:40,330 --> 01:11:44,420
If m is infinite, that becomes
an infinite number of bets.
1149
01:11:44,420 --> 01:11:48,095
One more theorem here-- this
one's a little surprising.
1150
01:11:52,172 --> 01:11:54,130
This theorem is called
Quit While You're Ahead.
1151
01:12:06,570 --> 01:12:18,700
If you start with n dollars,
and it's a fair game,
1152
01:12:18,700 --> 01:12:33,700
and you play until
you go broke, then
1153
01:12:33,700 --> 01:12:38,540
the probability that
you do go broke,
1154
01:12:38,540 --> 01:12:41,730
as opposed to playing
forever, is 1.
1155
01:12:41,730 --> 01:12:43,430
It's a certainty.
1156
01:12:43,430 --> 01:12:47,140
You'll go broke, even
though you expect it to take
1157
01:12:47,140 --> 01:12:49,570
an infinite amount of time.
1158
01:12:49,570 --> 01:12:51,074
All right, so let's prove that.
1159
01:13:06,561 --> 01:13:07,977
OK, the proof is
by contradiction.
1160
01:13:15,330 --> 01:13:18,600
Assume it's not true.
1161
01:13:18,600 --> 01:13:23,220
And that means that
you're assuming
1162
01:13:23,220 --> 01:13:26,080
that there exists
some number of dollars
1163
01:13:26,080 --> 01:13:32,880
that you can start with, and
some epsilon bigger than 0,
1164
01:13:32,880 --> 01:13:41,190
such that the probability
that you lose the n
1165
01:13:41,190 --> 01:13:43,900
dollars-- in which
case you're going home
1166
01:13:43,900 --> 01:13:56,310
broke-- let me write the
probability you go broke--
1167
01:13:56,310 --> 01:14:00,009
is at most 1 minus epsilon.
1168
01:14:00,009 --> 01:14:01,800
In other words, if the
theorem is not true,
1169
01:14:01,800 --> 01:14:04,860
there's some amount of money
you can start with such
1170
01:14:04,860 --> 01:14:09,000
that the chance you go broke
is less than 1-- less than 1
1171
01:14:09,000 --> 01:14:12,150
minus epsilon.
1172
01:14:12,150 --> 01:14:17,590
OK, now that means that for
all m, where you might possibly
1173
01:14:17,590 --> 01:14:20,680
stop but you're not
going to, the probability
1174
01:14:20,680 --> 01:14:31,365
you lose n before you win m
is at most 1 minus epsilon.
1175
01:14:35,700 --> 01:14:37,240
Because we're saying
the probability
1176
01:14:37,240 --> 01:14:39,360
you lose n no matter
what is at most that.
1177
01:14:39,360 --> 01:14:41,820
So it's certainly less
than 1 minus epsilon
1178
01:14:41,820 --> 01:14:46,670
that you lose n before
you win m dollars.
1179
01:14:46,670 --> 01:14:48,880
And we know what
that probability is.
1180
01:14:48,880 --> 01:14:51,770
This probability is
just m over n plus m.
1181
01:14:51,770 --> 01:14:54,700
We proved that earlier.
1182
01:14:54,700 --> 01:14:59,360
So that has to be less than
1 minus epsilon for all m.
1183
01:14:59,360 --> 01:15:05,150
And now I just multiply
through for all m.
1184
01:15:05,150 --> 01:15:07,570
That means that m is
less than or equal to 1
1185
01:15:07,570 --> 01:15:09,616
minus epsilon n plus m.
1186
01:15:12,930 --> 01:15:17,092
And then we'll solve that.
1187
01:15:25,960 --> 01:15:29,630
OK, so just multiply this out.
1188
01:15:29,630 --> 01:15:39,260
So for all m less than or
equal to n plus m minus epsilon
1189
01:15:39,260 --> 01:15:45,360
n minus epsilon m, and now
pull the m terms out here,
1190
01:15:45,360 --> 01:15:49,640
I get for all m,
epsilon m is less than
1191
01:15:49,640 --> 01:15:53,020
or equal to 1 minus epsilon n.
1192
01:15:53,020 --> 01:15:55,840
That means for all
m, m is smaller
1193
01:15:55,840 --> 01:16:01,170
than 1 minus epsilon
over epsilon times n.
1194
01:16:01,170 --> 01:16:03,060
And that can't be true.
1195
01:16:03,060 --> 01:16:06,270
It's not true the for all
m, this is less than that,
1196
01:16:06,270 --> 01:16:09,030
because these are fixed values.
1197
01:16:09,030 --> 01:16:11,790
That's a contradiction.
1198
01:16:11,790 --> 01:16:15,320
All right, so we
proved that if you
1199
01:16:15,320 --> 01:16:17,530
keep playing until
you're broke, you will go
1200
01:16:17,530 --> 01:16:20,030
broke with probability 1.
1201
01:16:20,030 --> 01:16:25,060
So even if you're playing a fair
game, quit while you're ahead.
1202
01:16:25,060 --> 01:16:28,970
Because if you don't,
you're going to go broke.
1203
01:16:28,970 --> 01:16:31,500
The swings will eventually
catch up with you.
1204
01:16:31,500 --> 01:16:35,159
So if we draw the graph here,
we'll see why that's true.
1205
01:16:51,810 --> 01:16:55,660
All right, if I have
time going this way,
1206
01:16:55,660 --> 01:17:01,750
and I start with n dollars,
my baseline is here.
1207
01:17:01,750 --> 01:17:03,884
The drift is 0.
1208
01:17:03,884 --> 01:17:04,925
I'm going to have swings.
1209
01:17:07,480 --> 01:17:09,510
I might have some
really big, high swings,
1210
01:17:09,510 --> 01:17:12,700
but it doesn't matter,
because eventually I'm
1211
01:17:12,700 --> 01:17:16,030
going to get a really bad swing,
and I'm going to go broke.
1212
01:17:19,450 --> 01:17:21,190
Now, if you ever
play a game where
1213
01:17:21,190 --> 01:17:25,960
you're likely to be winning each
time, and the drift goes up,
1214
01:17:25,960 --> 01:17:27,635
that's a good game
to play, obviously.
1215
01:17:27,635 --> 01:17:29,529
It just keeps getting better.
1216
01:17:29,529 --> 01:17:31,070
And that's a whole
math change there.
1217
01:17:34,410 --> 01:17:36,380
So that's it.
1218
01:17:36,380 --> 01:17:39,030
Remember, we have the ice
cream study session Monday.
1219
01:17:39,030 --> 01:17:41,790
So come to that if you'd like.
1220
01:17:41,790 --> 01:17:45,470
And definitely come to
the final on Tuesday.
1221
01:17:45,470 --> 01:17:47,440
And thanks for your
hard work, and being
1222
01:17:47,440 --> 01:17:49,170
such a great class this year.
1223
01:17:49,170 --> 01:17:52,820
[APPLAUSE]