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PROFESSOR: Now
today, we're going
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to talk about random walks.
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And in particular, we're going
to look at a classic phenomenon
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known as Gamblers Ruin.
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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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So it's actually a good review.
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We'll review recurrences.
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We'll review a lot
of probability laws.
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And it's actually a
nice problem to look at.
00:00:49.050 --> 00:00:52.090
It's another example where you
get a non-intuitive solution
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using probability.
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And if you like to
gamble, it's really good
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that you look at this problem
before you go to Vegas or down
00:00:59.330 --> 00:01:01.050
to Foxwoods.
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Now the Gambler's Ruin problem,
you start with n dollars.
00:01:19.330 --> 00:01:22.390
And we're going to do a
simplified version, where
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in each bet, you win
$1 or you lose $1.
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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It's more like $10.
00:01:31.150 --> 00:01:34.250
But just to make it
simple for counting,
00:01:34.250 --> 00:01:37.980
we're going to
assume that each bet
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you win $1 with
probability p, and you lose
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$1 with probability 1 minus p.
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And in this version,
we're going to assume
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you keep playing until
one of two things
00:02:00.610 --> 00:02:04.380
happens-- you get
ahead by m dollars,
00:02:04.380 --> 00:02:10.530
or you lose all the money you
came with-- all n dollars.
00:02:10.530 --> 00:02:19.850
So you play until you
win m more-- net m plus--
00:02:19.850 --> 00:02:23.780
or you lose n.
00:02:23.780 --> 00:02:25.030
And that's where you go broke.
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You run out of money.
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And we're going to assume
you don't borrow anything
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from the house.
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All right, and we're going
to look at the probability
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that you come out a winner
versus going home broke--
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that you made m dollars.
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Now, the game we're going
to analyze is roulette,
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but the technique
works for any of them.
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How many people
have played roulette
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before in some form or another?
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OK, so this is a
game where there's
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the ball that goes around the
dish, and you spin the wheel.
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And there's 36
numbers from 1 to 36.
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Half of them are
red, half are black.
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And then there's the zero and
the double zero that are green.
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And we're going to
look at the version
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where you just bet
on red or black.
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And you win if the ball
lands on a slot that's red.
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And there's 18 of those.
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And you lose otherwise.
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So in this case, the
probability of winning, p,
00:03:28.530 --> 00:03:31.520
is there's 18 chances to win.
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And it's not 36 total.
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It's 38 total because of the
zero and the double zero.
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All right so this is
9/19 chance of winning
00:03:42.680 --> 00:03:45.960
and a 10/19 chance of losing.
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And so this is a game that has a
chance of winning of about 47%,
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so it's almost a fair game.
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It's not 50-50.
00:03:54.869 --> 00:03:57.160
And that's because the casino's
got to make some money.
00:03:57.160 --> 00:03:59.400
I mean, they have
the big facility.
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They're giving you free
drinks, and all the rest.
00:04:02.010 --> 00:04:03.880
So they got to
make money somehow.
00:04:03.880 --> 00:04:07.440
And they make money on
this bet because they're
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going to make $0.03
on the dollar here.
00:04:12.430 --> 00:04:13.780
You're going to wager.
00:04:13.780 --> 00:04:18.230
And then you're going
to come back with 47%.
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And people generally
are fine with that.
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They don't expect to have
the odds in their favor
00:04:22.900 --> 00:04:26.010
when you're gambling
in a casino.
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Now, in an effort to sort
of come home a winner,
00:04:30.180 --> 00:04:32.780
the way people do that--
knowing that the odds are
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a little against
them-- is they might
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put more money in
their pocket coming in
00:04:38.580 --> 00:04:40.050
than they expect to win.
00:04:40.050 --> 00:04:43.760
So often, you'll see
people come into the casino
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with the goal of winning 100,
but they start with 1,000
00:04:48.829 --> 00:04:49.495
in their pocket.
00:04:52.730 --> 00:04:55.440
So they're willing
to risk $1,000,
00:04:55.440 --> 00:04:59.790
but they're going to quit
happy if they get up 100.
00:04:59.790 --> 00:05:03.450
OK so you either go
home with $1,100,
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or you're going home
with $0, in this case.
00:05:05.550 --> 00:05:07.760
And you came with $1,000.
00:05:07.760 --> 00:05:10.300
And this means that you're--
at least the thinking goes--
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this means you're more
likely to go home happy.
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If you quit when
you get up by 100,
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you're more likely
to land there,
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because it's almost a fair game,
than you are to lose all 1,000.
00:05:21.650 --> 00:05:23.570
That's the thinking anyway.
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In fact, my mother-in-law
plays roulette, red and black,
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and she follows the strategy.
00:05:30.040 --> 00:05:32.570
And she claims that she
does this for that reason--
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that she almost always wins.
00:05:35.510 --> 00:05:37.820
She goes home happy
almost always.
00:05:37.820 --> 00:05:40.290
And that's the
important thing here.
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And it does reasonable,
because after all, roulette
00:05:42.500 --> 00:05:45.980
is almost a fair game.
00:05:45.980 --> 00:05:47.470
So what do you think?
00:05:47.470 --> 00:05:49.100
How many people
think she's right
00:05:49.100 --> 00:05:52.650
that she almost always wins?
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Anybody?
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I have sort of set it up.
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It's my mother-in-law,
after all,
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so probably she's
going to be wrong.
00:06:01.740 --> 00:06:06.480
Well, how many people think
it's better than a 50% chance
00:06:06.480 --> 00:06:11.610
you win $100 before
you lose $1,000?
00:06:11.610 --> 00:06:15.030
That's probably
more-- how many people
00:06:15.030 --> 00:06:19.350
think you're more likely to
lose $1,000 before you win $100?
00:06:19.350 --> 00:06:21.960
Wow, OK, so you've been
to 6.04 too long now.
00:06:21.960 --> 00:06:24.420
OK, what about this-- how
many people think you're
00:06:24.420 --> 00:06:30.505
more likely to lose
$10,000 than to win $100?
00:06:30.505 --> 00:06:32.130
All right, how many
people think you're
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more likely to lose $1 million?
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A bunch of you still think that.
00:06:38.030 --> 00:06:40.370
OK, well, you're right.
00:06:40.370 --> 00:06:45.830
In fact, it is almost
certain you will go broke,
00:06:45.830 --> 00:06:51.730
no matter how much money you
bring, before you win $100.
00:06:51.730 --> 00:06:55.290
In fact, we're
going to prove today
00:06:55.290 --> 00:07:02.080
that the probability that
you win $100 before losing
00:07:02.080 --> 00:07:04.870
$100 million if you
stayed long enough-- that
00:07:04.870 --> 00:07:09.930
takes a while-- the chance you
go home a winner is less than 1
00:07:09.930 --> 00:07:14.470
in 37,648.
00:07:14.470 --> 00:07:18.326
You have no chance
to go home happy.
00:07:18.326 --> 00:07:20.950
So my mother-in-law's telling me
the story about how she always
00:07:20.950 --> 00:07:21.740
goes home happy.
00:07:21.740 --> 00:07:24.960
And I'm saying, no, no,
wait a minute, you can't.
00:07:24.960 --> 00:07:26.230
You never went home happy.
00:07:26.230 --> 00:07:28.040
Let's be honest.
00:07:28.040 --> 00:07:29.057
It can't be.
00:07:29.057 --> 00:07:30.390
She goes, no, no, no, it's true.
00:07:30.390 --> 00:07:32.905
I go, no, look, there's
a mathematical proof.
00:07:32.905 --> 00:07:33.530
I have a proof.
00:07:33.530 --> 00:07:38.140
I can show you my proof-- very
unlikely you go home a winner.
00:07:38.140 --> 00:07:40.150
So somehow, she's
not very impressed
00:07:40.150 --> 00:07:42.300
with the mathematical proof.
00:07:42.300 --> 00:07:43.320
And she keeps insisting.
00:07:43.320 --> 00:07:45.400
And I keep trying to
show her the proof.
00:07:45.400 --> 00:07:48.170
And anyway, I hope I'll have
more luck with you guys today
00:07:48.170 --> 00:07:51.520
in showing you the proof that
the chance you go home happy
00:07:51.520 --> 00:07:54.600
here is very, very small.
00:07:54.600 --> 00:07:56.240
Now, in the end, I
didn't convince her,
00:07:56.240 --> 00:07:59.680
but we'll see how
we do here today.
00:07:59.680 --> 00:08:02.940
Now, in order to see why this
probability is so stunningly
00:08:02.940 --> 00:08:06.690
small-- you would just
never guess it's that low--
00:08:06.690 --> 00:08:09.510
we've got to learn
about random walks.
00:08:09.510 --> 00:08:12.010
And they come up in all
sorts of applications.
00:08:12.010 --> 00:08:16.330
In fact, page rank--
that got Google started--
00:08:16.330 --> 00:08:19.390
it's all based on a random
walk through the Web
00:08:19.390 --> 00:08:24.600
or through the links on
web pages on that graph.
00:08:24.600 --> 00:08:26.380
Now, for the gambling
problem, we're
00:08:26.380 --> 00:08:28.180
going to look at a
very special case--
00:08:28.180 --> 00:08:30.210
probably the simplest
case of a random walk--
00:08:30.210 --> 00:08:33.390
and that's a
one-dimensional random walk.
00:08:33.390 --> 00:08:35.049
In a one-dimensional
random walk,
00:08:35.049 --> 00:08:37.809
there's some value-- say
the number of dollars
00:08:37.809 --> 00:08:40.240
you've got in your pocket.
00:08:40.240 --> 00:08:43.309
And this value can
go up, or go down,
00:08:43.309 --> 00:08:47.310
or stay the same each time you
do something like make a bet.
00:08:47.310 --> 00:08:51.760
And each of this happens
with a certain probability.
00:08:51.760 --> 00:08:55.360
Now in this case, you
either go up by one,
00:08:55.360 --> 00:08:58.010
or you go down by one, and
you can't stay the same.
00:08:58.010 --> 00:09:00.100
Every bet you win
$1 or you lose $1.
00:09:00.100 --> 00:09:02.490
So it's really a special case.
00:09:02.490 --> 00:09:05.700
And we can diagram
it as follows.
00:09:05.700 --> 00:09:09.700
We can put time, or the
number of bets, on this axis.
00:09:12.900 --> 00:09:16.365
And we can put the number
of dollars on this axis.
00:09:20.120 --> 00:09:25.010
Now in this case, we
start with n dollars.
00:09:25.010 --> 00:09:28.990
And we might win the first
bet, so we go to n plus 1.
00:09:28.990 --> 00:09:35.240
We might lose a bet, might lose
again, could win the next one,
00:09:35.240 --> 00:09:39.770
lose, win, lose, lose.
00:09:39.770 --> 00:09:45.210
So this corresponds to a
string-- win, lose, lose, lose,
00:09:45.210 --> 00:09:53.290
lose, win, lose, win,
lose, lose, lose.
00:09:53.290 --> 00:09:57.180
And when we win, we go up $1.
00:09:57.180 --> 00:10:01.260
When we lose, we go down $1.
00:10:01.260 --> 00:10:03.260
And it's called
one-dimensional, because there's
00:10:03.260 --> 00:10:05.050
just one thing that's changing.
00:10:05.050 --> 00:10:07.510
You're going up and down there.
00:10:07.510 --> 00:10:11.420
Now, the probability
of going up is p.
00:10:11.420 --> 00:10:13.990
And that's no matter
what happened before.
00:10:13.990 --> 00:10:17.380
It's a memoryless
independent system.
00:10:17.380 --> 00:10:20.950
The probability you win your
i-th bet has nothing to do--
00:10:20.950 --> 00:10:23.380
is totally independent,
mutually independent-- of all
00:10:23.380 --> 00:10:26.900
the other bets that
took place before.
00:10:26.900 --> 00:10:28.770
So let's write that down.
00:10:32.410 --> 00:10:40.040
So the probability
of an up move is p.
00:10:40.040 --> 00:10:46.210
The probability of a
down move is 1 minus p.
00:10:46.210 --> 00:10:52.720
And these are mutually
independent of past moves.
00:10:56.840 --> 00:10:59.080
Now, when you have
a random walk where
00:10:59.080 --> 00:11:01.060
the moves are
mutually independent,
00:11:01.060 --> 00:11:02.739
it has a special name.
00:11:02.739 --> 00:11:03.780
It's called a martingale.
00:11:10.120 --> 00:11:12.070
All random walks
don't have to have
00:11:12.070 --> 00:11:13.539
mutually independent steps.
00:11:13.539 --> 00:11:15.330
Say you're looking
about winning and losing
00:11:15.330 --> 00:11:17.422
a baseball game in a series.
00:11:17.422 --> 00:11:19.630
We looked at a scenario
where, if you lost yesterday,
00:11:19.630 --> 00:11:22.310
you're feeling lousy,
more likely to lose today.
00:11:22.310 --> 00:11:24.140
Not true in the
gambling case here.
00:11:24.140 --> 00:11:25.261
It's mutually independent.
00:11:25.261 --> 00:11:26.760
And that's the only
case we're going
00:11:26.760 --> 00:11:29.290
to study for random walks.
00:11:29.290 --> 00:11:36.585
Now, if p is not 1/2, the random
walk is said to be biased.
00:11:43.720 --> 00:11:45.430
And that's what
happens in the casino.
00:11:45.430 --> 00:11:49.040
It's biased in
favor of the house.
00:11:49.040 --> 00:11:53.460
If p equals 1/2, then the
random walk is unbiased.
00:12:00.440 --> 00:12:03.360
Now, in this particular
case that we're looking at,
00:12:03.360 --> 00:12:07.070
we have boundaries
on the random walk.
00:12:07.070 --> 00:12:10.000
There's a boundary
at 0, because you
00:12:10.000 --> 00:12:12.220
go home broke if
you lost everything.
00:12:12.220 --> 00:12:15.770
If the random walk ever
hit $0, you're done.
00:12:15.770 --> 00:12:23.100
And we're also going to
put a boundary at n plus m.
00:12:26.220 --> 00:12:30.690
So I'm going to have
a boundary here.
00:12:30.690 --> 00:12:37.180
So that if I win m dollars here,
I stop and I go home happy.
00:12:37.180 --> 00:12:41.790
If the random walk ever
goes here, then I stop.
00:12:41.790 --> 00:12:45.970
Those are called boundary
conditions for the walk.
00:12:45.970 --> 00:12:49.800
And what we want to do is
analyze the probability
00:12:49.800 --> 00:12:52.940
that we hit that top
boundary before we
00:12:52.940 --> 00:12:55.630
hit the bottom boundary.
00:12:55.630 --> 00:12:58.000
So we're going to define
that event to be W star.
00:13:00.580 --> 00:13:10.410
W star is the event that the
random walk hits T, which
00:13:10.410 --> 00:13:17.680
is n plus m, before it hits 0.
00:13:21.240 --> 00:13:26.080
In other words, you go home
happy without going broke.
00:13:26.080 --> 00:13:32.020
Let's also define D to be the
number of dollars at the start.
00:13:32.020 --> 00:13:34.930
And this is just going
to be n in our case.
00:13:40.960 --> 00:13:48.300
We're interested in, call it
X sub n is the probability
00:13:48.300 --> 00:13:53.750
that we go home happy given
we started with n dollars.
00:13:56.330 --> 00:13:57.740
And that's a function of n.
00:13:57.740 --> 00:14:00.650
So we'll make a
variable called X n.
00:14:00.650 --> 00:14:02.710
And we want to know what
that probability is.
00:14:02.710 --> 00:14:04.293
And of course, the
more you come with,
00:14:04.293 --> 00:14:07.775
you'd think it's a
higher chance of winning
00:14:07.775 --> 00:14:09.150
the more you have
in your pocket,
00:14:09.150 --> 00:14:11.250
because you can play for more.
00:14:11.250 --> 00:14:14.570
So the goal is to
figure this out.
00:14:14.570 --> 00:14:18.050
Now to do this, we could
use the tree method.
00:14:18.050 --> 00:14:22.110
But it gets pretty complicated,
because the sample space
00:14:22.110 --> 00:14:26.390
is the sample space of
all one-loss sequences.
00:14:26.390 --> 00:14:28.130
And how big is
that sample space?
00:14:32.106 --> 00:14:33.100
AUDIENCE: Infinite.
00:14:33.100 --> 00:14:35.050
PROFESSOR: Infinite.
00:14:35.050 --> 00:14:36.977
I could play forever.
00:14:36.977 --> 00:14:39.560
All right, now it turns out the
probability of playing forever
00:14:39.560 --> 00:14:40.080
is 0.
00:14:40.080 --> 00:14:43.350
And we won't prove that, but
there are an infinite number
00:14:43.350 --> 00:14:44.150
of sample points.
00:14:44.150 --> 00:14:46.150
So doing the tree method
is a little complicated
00:14:46.150 --> 00:14:49.060
when it's infinite.
00:14:49.060 --> 00:14:52.280
So what we're going
to do is use some
00:14:52.280 --> 00:14:54.950
of the theorems we've proved
over the last few weeks
00:14:54.950 --> 00:14:57.835
and set up a recurrence
to find this probability.
00:15:01.100 --> 00:15:04.640
Now, I'm going to tell you
what the recurrence is,
00:15:04.640 --> 00:15:09.730
and then prove
that that's right.
00:15:09.730 --> 00:15:17.580
So I claim that X
n is 0 probability
00:15:17.580 --> 00:15:21.320
if we start with $0.
00:15:21.320 --> 00:15:25.730
It's 1 if we start
with T dollars.
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
00:15:36.580 --> 00:15:43.100
if we start with between
$0 and T dollars.
00:15:47.110 --> 00:15:49.450
All right, so that's
what I claim X n is.
00:15:49.450 --> 00:15:51.960
And it's, of course, a
recursion that I've set up here.
00:15:51.960 --> 00:15:53.650
So let's see why
that's the case.
00:15:58.230 --> 00:16:00.930
OK, so let's check the 0 case.
00:16:00.930 --> 00:16:07.530
X 0 is the probability
we go home a winner given
00:16:07.530 --> 00:16:09.210
we started with $0.
00:16:12.510 --> 00:16:13.475
Why is that 0?
00:16:16.974 --> 00:16:17.890
AUDIENCE: [INAUDIBLE].
00:16:17.890 --> 00:16:19.386
PROFESSOR: What's that?
00:16:19.386 --> 00:16:21.180
AUDIENCE: [INAUDIBLE].
00:16:21.180 --> 00:16:23.280
PROFESSOR: Yeah,
you started broke.
00:16:23.280 --> 00:16:25.520
You never get off the
ground, because you
00:16:25.520 --> 00:16:27.730
quit as soon as you have $0.
00:16:27.730 --> 00:16:35.020
So you have no chance to win,
because you're broke to start.
00:16:35.020 --> 00:16:41.670
Let's check the next case,
X T-- case n equals T--
00:16:41.670 --> 00:16:45.570
is the probability you
go home a winner given
00:16:45.570 --> 00:16:49.310
you started with T dollars.
00:16:49.310 --> 00:16:51.750
Why is that 1?
00:16:51.750 --> 00:16:55.147
Why is that certain, sort
of from the definition?
00:16:55.147 --> 00:16:56.580
AUDIENCE: [INAUDIBLE].
00:16:56.580 --> 00:16:58.230
PROFESSOR: You already
have your money.
00:16:58.230 --> 00:16:59.560
You already hit
the top boundary,
00:16:59.560 --> 00:17:00.643
because you started there.
00:17:00.643 --> 00:17:02.440
Remember, you quit
and you're happy.
00:17:02.440 --> 00:17:05.900
Go home happy if
you hit T dollars.
00:17:05.900 --> 00:17:08.697
All right, so you're
guaranteed to go home happy,
00:17:08.697 --> 00:17:10.030
because you never make any bets.
00:17:10.030 --> 00:17:13.530
You started with all the money
you needed to go home happy.
00:17:13.530 --> 00:17:18.490
Then we have the
interesting case,
00:17:18.490 --> 00:17:20.849
where you start with
between 0 and T dollars.
00:17:20.849 --> 00:17:23.990
And now you're going
to make some bets.
00:17:23.990 --> 00:17:26.310
And then X n is the
probability-- just
00:17:26.310 --> 00:17:30.590
the definition-- of going
home happy-- i.e. winning
00:17:30.590 --> 00:17:34.680
and having T dollars,
if you start with n.
00:17:34.680 --> 00:17:38.710
Now, there's two cases to
analyze this, based on what
00:17:38.710 --> 00:17:40.380
happens in the first bet.
00:17:40.380 --> 00:17:43.600
You could win it, or
you could lose it.
00:17:43.600 --> 00:17:46.240
And then we're going to recurse.
00:17:46.240 --> 00:17:54.840
So we're going to define
E to be the event that you
00:17:54.840 --> 00:17:57.340
win the first bet.
00:18:00.570 --> 00:18:07.520
And E bar is the event that
you lose the first bet.
00:18:12.080 --> 00:18:14.340
Now, by the theory of
total probability, which
00:18:14.340 --> 00:18:18.120
we did in recitation
maybe a couple weeks ago,
00:18:18.120 --> 00:18:21.970
we can rewrite this depending
on whether E happened
00:18:21.970 --> 00:18:25.080
or the complement of E happened.
00:18:25.080 --> 00:18:28.370
And you get that the
probability is simply
00:18:28.370 --> 00:18:31.990
the probability of
going home happy
00:18:31.990 --> 00:18:34.844
and winning the
first bet times--
00:18:34.844 --> 00:18:36.510
and I've got to put
the conditioning in.
00:18:36.510 --> 00:18:37.580
That doesn't go away.
00:18:44.080 --> 00:18:45.510
So I'm breaking into two cases.
00:18:45.510 --> 00:18:51.350
The first one is you win the
first bet given D equals n,
00:18:51.350 --> 00:18:59.470
And the case where you lose the
first bet, given D equals n.
00:19:02.070 --> 00:19:04.601
Any questions here?
00:19:04.601 --> 00:19:06.530
The probability of
going home happy
00:19:06.530 --> 00:19:09.660
given you start with n
dollars is the probability
00:19:09.660 --> 00:19:17.050
of going home happy and winning
the first bet given D equals
00:19:17.050 --> 00:19:19.465
n plus the probability
of going home happy
00:19:19.465 --> 00:19:22.800
and losing the first
bet given D equals n--
00:19:22.800 --> 00:19:25.630
just those are the two cases.
00:19:25.630 --> 00:19:29.290
Now I can use the definition
of conditional probability
00:19:29.290 --> 00:19:31.256
to rewrite these.
00:19:31.256 --> 00:19:34.350
This is the probability--
you've got two events--
00:19:34.350 --> 00:19:39.520
that the first one
happens given D equals n
00:19:39.520 --> 00:19:43.440
times the probability
the second one happens
00:19:43.440 --> 00:19:49.180
given that the first one
happened and D equals n.
00:19:49.180 --> 00:19:51.560
This is just the definition
of conditional probability,
00:19:51.560 --> 00:19:54.180
when I've got an
intersection of events here.
00:19:54.180 --> 00:19:55.870
The probability
of both happening
00:19:55.870 --> 00:19:57.760
is the probability of
the first happening
00:19:57.760 --> 00:19:59.884
times the probability of
the second happening given
00:19:59.884 --> 00:20:00.960
that the first happened.
00:20:00.960 --> 00:20:05.570
And of course, everything is
in this universe of D equals n.
00:20:05.570 --> 00:20:07.670
So I've used it in a
little different twist
00:20:07.670 --> 00:20:09.490
than we had it before.
00:20:09.490 --> 00:20:11.240
The same thing over
here-- this now
00:20:11.240 --> 00:20:17.140
is the probability of E
prime given D equals n
00:20:17.140 --> 00:20:20.390
times the probability of
W star-- winning, going
00:20:20.390 --> 00:20:29.420
home happy-- given that you lost
the first bet and D equals n.
00:20:29.420 --> 00:20:32.370
That's D equals n there.
00:20:32.370 --> 00:20:34.540
So it looks like it's
got more complicated,
00:20:34.540 --> 00:20:36.005
but now we can
start simplifying.
00:20:38.650 --> 00:20:40.610
What's the
probability of winning
00:20:40.610 --> 00:20:44.022
the first bet given that
you started with n dollars?
00:20:44.022 --> 00:20:45.390
AUDIENCE: p.
00:20:45.390 --> 00:20:47.565
PROFESSOR: p-- in
fact, does this
00:20:47.565 --> 00:20:49.690
have anything to do with
the probability of winning
00:20:49.690 --> 00:20:51.730
the first bet?
00:20:51.730 --> 00:20:53.870
No, this is just p.
00:20:59.610 --> 00:21:03.290
Now, what about this thing?
00:21:03.290 --> 00:21:13.620
I am conditioning on
winning the first bet given
00:21:13.620 --> 00:21:15.355
and I start with n dollars.
00:21:19.750 --> 00:21:22.930
What's another way of
expressing I won the first bet
00:21:22.930 --> 00:21:25.310
and I started with n dollars?
00:21:25.310 --> 00:21:26.087
Yeah?
00:21:26.087 --> 00:21:27.520
AUDIENCE: You have n plus $1.
00:21:27.520 --> 00:21:31.860
PROFESSOR: I now have n
plus $1 going forward.
00:21:31.860 --> 00:21:34.900
And because I have a
martingale, and everything
00:21:34.900 --> 00:21:37.340
is mutually independent,
it's like the world
00:21:37.340 --> 00:21:38.700
starts all over again.
00:21:38.700 --> 00:21:41.440
I'm now in a state
with n plus $1,
00:21:41.440 --> 00:21:44.110
and I want to know
the probability
00:21:44.110 --> 00:21:45.375
that I go home happy.
00:21:45.375 --> 00:21:49.000
It doesn't matter how
I got the n plus $1.
00:21:49.000 --> 00:21:52.000
It's just going forward-- I
got n plus $1 in my pocket,
00:21:52.000 --> 00:21:54.270
I want to know the probability
of going home happy.
00:21:54.270 --> 00:21:57.040
So I reset to D equals n plus 1.
00:21:59.740 --> 00:22:03.892
So I replace this with that,
because however long it
00:22:03.892 --> 00:22:05.850
took me to get there and
all that stuff doesn't
00:22:05.850 --> 00:22:07.660
matter for this analysis.
00:22:07.660 --> 00:22:08.990
It's all mutually dependent.
00:22:13.110 --> 00:22:16.360
Probability of losing the
first bet given that I started
00:22:16.360 --> 00:22:20.490
with n dollars-- 1 minus p.
00:22:20.490 --> 00:22:23.290
Doesn't matter how
much I started with.
00:22:23.290 --> 00:22:29.120
And here, I want to know the
probability of going home happy
00:22:29.120 --> 00:22:31.950
given-- well, if I
lost the first bet
00:22:31.950 --> 00:22:36.080
and I started with
n, what have I got?
00:22:36.080 --> 00:22:37.087
n minus 1.
00:22:40.530 --> 00:22:44.746
It doesn't matter how
I got to n minus 1.
00:22:44.746 --> 00:22:46.370
Now this is going to
get really simple.
00:22:49.250 --> 00:22:52.720
What's another name
for that expression?
00:22:52.720 --> 00:22:53.830
X n plus 1.
00:22:57.560 --> 00:23:00.800
And another name
for this expression?
00:23:00.800 --> 00:23:02.003
X n minus 1.
00:23:05.680 --> 00:23:11.720
So we proved that X n equals
p X n plus 1 plus 1 minus
00:23:11.720 --> 00:23:12.770
p X n minus 1.
00:23:12.770 --> 00:23:15.100
And that's what I
claimed is true.
00:23:15.100 --> 00:23:18.380
So we finished the proof.
00:23:18.380 --> 00:23:20.214
Any questions?
00:23:20.214 --> 00:23:21.130
AUDIENCE: [INAUDIBLE].
00:23:21.130 --> 00:23:22.356
PROFESSOR: Did i screw it up?
00:23:22.356 --> 00:23:23.272
AUDIENCE: [INAUDIBLE].
00:23:23.272 --> 00:23:29.940
PROFESSOR: I claim probability
of winning-- so let's
00:23:29.940 --> 00:23:31.590
see if I have a wrong in here.
00:23:31.590 --> 00:23:32.950
I might have screwed it up.
00:23:32.950 --> 00:23:38.440
I think I proved
it's n plus 1, right?
00:23:38.440 --> 00:23:41.230
Yep, sure enough, I
think this is a plus 1.
00:23:41.230 --> 00:23:43.210
That's a minus 1.
00:23:43.210 --> 00:23:45.692
Now, it's always good to
check to you proved what you
00:23:45.692 --> 00:23:46.900
said you were going to prove.
00:23:46.900 --> 00:23:50.006
So I needed to change this.
00:23:50.006 --> 00:23:50.880
That's what I proved.
00:23:57.290 --> 00:23:58.239
Any other questions?
00:23:58.239 --> 00:23:59.780
That was a pretty
important question.
00:24:02.370 --> 00:24:07.620
All right, so we have
a recurrence for X n.
00:24:07.620 --> 00:24:10.430
Now, it's a little
funny looking at first,
00:24:10.430 --> 00:24:12.430
because normally
with a recurrence,
00:24:12.430 --> 00:24:15.770
X n would depend on X
sub i that are smaller--
00:24:15.770 --> 00:24:18.040
the i's are smaller than n.
00:24:18.040 --> 00:24:20.210
So it looks a little wacky.
00:24:20.210 --> 00:24:21.470
But is that a problem?
00:24:24.320 --> 00:24:26.550
I can just solve
for X n plus 1--
00:24:26.550 --> 00:24:29.550
just subtract this
and put it over there.
00:24:29.550 --> 00:24:30.540
So let's do that.
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
00:24:53.820 --> 00:25:03.210
on its own side, I get p X
n plus 1 minus X n plus 1
00:25:03.210 --> 00:25:08.690
minus p X n minus 1 equals 0.
00:25:08.690 --> 00:25:11.260
And I know that X 0 is 0.
00:25:11.260 --> 00:25:13.445
And I know that X T equals 1.
00:25:17.314 --> 00:25:18.855
Now, what type of
recurrence is this?
00:25:21.860 --> 00:25:22.770
AUDIENCE: Linear.
00:25:22.770 --> 00:25:26.130
PROFESSOR: Linear, good, so
it's a linear recurrence.
00:25:26.130 --> 00:25:28.853
And what type of linear
recurrence is it?
00:25:28.853 --> 00:25:29.800
AUDIENCE: Homogeneous.
00:25:29.800 --> 00:25:33.200
PROFESSOR: Homogeneous-- that's
the best case, simple case,
00:25:33.200 --> 00:25:34.039
that's good.
00:25:34.039 --> 00:25:35.830
The boundary conditions
are a little weird,
00:25:35.830 --> 00:25:38.820
because the recurrences
we all saw before,
00:25:38.820 --> 00:25:40.320
if we had two
boundary conditions it
00:25:40.320 --> 00:25:41.870
would be X0 and X1.
00:25:41.870 --> 00:25:43.260
Here it's X0 and X T.
00:25:43.260 --> 00:25:45.151
But all's you need are two.
00:25:45.151 --> 00:25:46.400
Doesn't matter where they are.
00:25:48.960 --> 00:25:50.220
So how do I solve that thing?
00:25:50.220 --> 00:25:54.070
What's the next thing I do?
00:25:54.070 --> 00:25:55.041
What is it?
00:25:55.041 --> 00:25:56.540
AUDIENCE: Characterize
the equation.
00:25:56.540 --> 00:25:57.560
PROFESSOR: Characterize
the equation.
00:25:57.560 --> 00:25:59.294
And what do you do
it that equation?
00:25:59.294 --> 00:26:00.210
AUDIENCE: [INAUDIBLE].
00:26:00.210 --> 00:26:01.668
PROFESSOR: Solve
it, get the roots.
00:26:01.668 --> 00:26:03.820
This'll be good
practice for the final,
00:26:03.820 --> 00:26:06.797
because you'll probably have
to do something like this.
00:26:06.797 --> 00:26:08.380
So that's the
characteristic equation.
00:26:12.800 --> 00:26:15.940
And what's the order of
this equation-- the degree?
00:26:20.340 --> 00:26:26.730
That's going to be 2, right?
00:26:26.730 --> 00:26:33.600
I'm going to have pr squared
minus r plus 1 minus p is 0.
00:26:33.600 --> 00:26:37.177
That's my
characteristic equation.
00:26:37.177 --> 00:26:37.760
Remember that?
00:26:37.760 --> 00:26:40.640
So I make this be
the constant term.
00:26:40.640 --> 00:26:42.420
Then I have the
first-order term, then
00:26:42.420 --> 00:26:45.210
the second-order term.
00:26:45.210 --> 00:26:48.570
All right, now I solve it.
00:26:48.570 --> 00:26:52.080
And that's easy for a
second-order equation.
00:26:52.080 --> 00:26:59.540
1 plus or minus the square
root of 1 minus 4p 1
00:26:59.540 --> 00:27:03.900
minus p over 2 p.
00:27:07.140 --> 00:27:08.030
Let's do that.
00:27:10.810 --> 00:27:15.380
OK, so this is 1 plus
or minus the square root
00:27:15.380 --> 00:27:20.460
of 1 minus 4p plus
4p squared over 2p.
00:27:28.689 --> 00:27:30.730
Just using the quadratic
formula and simplifying.
00:27:35.780 --> 00:27:38.930
And it works out really
nicely, because that
00:27:38.930 --> 00:27:42.410
is the square root of-- this
is just 1 minus 2p squared.
00:27:48.990 --> 00:27:55.740
So that's 1 plus or
minus 1 minus 2p over 2p.
00:27:55.740 --> 00:28:06.410
And that is 2 minus 2p over
2p or 1 minus 1 cancels,
00:28:06.410 --> 00:28:13.020
then minus 2p is 2p over 2p.
00:28:13.020 --> 00:28:17.590
So the answers, the roots
are divide by 2 on this one.
00:28:17.590 --> 00:28:22.947
I get 1 minus p over p and 1.
00:28:22.947 --> 00:28:23.780
Those are the roots.
00:28:27.582 --> 00:28:28.665
Are these roots different?
00:28:30.960 --> 00:28:32.460
Do I have the case
of a double root?
00:28:32.460 --> 00:28:36.407
Are the roots always different?
00:28:36.407 --> 00:28:37.490
They're usually different.
00:28:37.490 --> 00:28:39.880
What's the case where
these roots are the same?
00:28:39.880 --> 00:28:40.870
AUDIENCE: 0.5.
00:28:40.870 --> 00:28:43.760
PROFESSOR: 0.5, which is
sort of an interesting case
00:28:43.760 --> 00:28:45.990
in this game.
00:28:45.990 --> 00:28:49.110
Because if p equals 1/2, we
have an unbiased random walk.
00:28:49.110 --> 00:28:51.250
You got a fair game.
00:28:51.250 --> 00:28:53.730
And so it says right away,
well, maybe the result
00:28:53.730 --> 00:28:56.807
is going to be different for
a fair game than the game
00:28:56.807 --> 00:28:58.765
we're playing in the
casino, where it's biased.
00:29:01.300 --> 00:29:04.432
So let's look at the casino
game where p is not 1/2.
00:29:04.432 --> 00:29:05.640
Then the roots are different.
00:29:08.280 --> 00:29:10.000
Later, we'll go back
and analyze the case
00:29:10.000 --> 00:29:13.053
when the roots of the
same for the fair game.
00:29:20.310 --> 00:29:30.250
So if p is not 1/2, then
we can solve for X n.
00:29:30.250 --> 00:29:37.960
X n is some constant times the
first root to the nth power
00:29:37.960 --> 00:29:42.180
plus a constant times the
second root to the nth power.
00:29:42.180 --> 00:29:45.515
Remember, that's how it works
for any linear homogeneous
00:29:45.515 --> 00:29:46.015
recurrence.
00:29:49.300 --> 00:29:53.900
And that's easy, because
the second root was 1.
00:29:53.900 --> 00:29:57.200
This is just plus
B. 1 to the n is 1.
00:29:59.960 --> 00:30:03.050
How do I figure out
what A and B are?
00:30:03.050 --> 00:30:04.300
AUDIENCE: Boundary conditions.
00:30:04.300 --> 00:30:07.090
PROFESSOR: Boundary
conditions, very good.
00:30:07.090 --> 00:30:08.855
So let's look at the
boundary conditions.
00:30:30.250 --> 00:30:36.100
OK, so the first boundary
condition is at 0.
00:30:36.100 --> 00:30:40.410
So we have 0 equals X 0.
00:30:40.410 --> 00:30:44.570
Plugging in there-- oops
I forgot the n up here.
00:30:44.570 --> 00:30:48.140
Plugging in n equals 0--
well, this to the 0 is just 1.
00:30:48.140 --> 00:30:57.054
That is A plus B. That
means that B equals minus A.
00:30:57.054 --> 00:30:58.470
Then the second
boundary condition
00:30:58.470 --> 00:31:05.880
is 1 equals X sub T.
And that is A 1 minus
00:31:05.880 --> 00:31:13.442
p over p to the T plus
B, but B was minus A.
00:31:13.442 --> 00:31:16.860
And now I can solve for A.
00:31:16.860 --> 00:31:28.850
So that means that A equals
1 over 1 minus p over p
00:31:28.850 --> 00:31:32.200
to the T minus 1.
00:31:32.200 --> 00:31:38.610
And B is negative A--
minus 1 over 1 minus p,
00:31:38.610 --> 00:31:40.810
over p to the T minus 1.
00:31:43.540 --> 00:31:51.140
And then I plug those back
in to the formula for X n.
00:31:51.140 --> 00:31:54.320
So here's my constant A.
I multiply that times 1
00:31:54.320 --> 00:31:59.020
minus p over p to the
n, plus I add this in.
00:31:59.020 --> 00:32:02.620
So this means that the
probability of going home
00:32:02.620 --> 00:32:10.500
a winner is 1 minus p over p to
the n over that thing-- 1 minus
00:32:10.500 --> 00:32:14.830
p over p to the T
minus 1, plus the B
00:32:14.830 --> 00:32:22.910
term, which really is a minus
term here, is just minus 1.
00:32:22.910 --> 00:32:23.880
Put that on top here.
00:32:26.620 --> 00:32:30.350
That sort of looks
messy, but there's
00:32:30.350 --> 00:32:34.780
a simplification to get an
upper bound that's very close.
00:32:34.780 --> 00:32:38.870
In particular, if you have
a biased game against you--
00:32:38.870 --> 00:32:43.960
so if p is less than 1/2,
as it is in roulette,
00:32:43.960 --> 00:32:47.720
then this is a
number bigger than 1.
00:32:47.720 --> 00:32:53.105
That means that 1 minus p
over p is bigger than 1.
00:32:56.340 --> 00:32:57.990
So this is bigger than 1.
00:32:57.990 --> 00:32:58.990
This is bigger than 1.
00:32:58.990 --> 00:33:01.620
T is the upper limit.
00:33:01.620 --> 00:33:03.410
It's n plus m.
00:33:03.410 --> 00:33:05.970
So I've got a bigger number
down here than I do here.
00:33:05.970 --> 00:33:08.750
So overall, it's a
fraction less than 1.
00:33:08.750 --> 00:33:10.860
And when you have a
fraction less than 1,
00:33:10.860 --> 00:33:14.330
if you add 1 to the
numerator and denominator,
00:33:14.330 --> 00:33:15.480
it gets closer to 1.
00:33:15.480 --> 00:33:17.670
It gets bigger.
00:33:17.670 --> 00:33:23.460
So this is upper-bounded by
just adding 1 to each of these.
00:33:23.460 --> 00:33:28.440
Its upper-bounded by this
over that, which is 1 minus p
00:33:28.440 --> 00:33:36.860
over p to the n minus T.
And T is just n plus m.
00:33:36.860 --> 00:33:40.930
So this equals-- why don't
I turn it upside down?
00:33:40.930 --> 00:33:44.720
Make it p over 1 minus p to get
a fraction that's less than 1.
00:33:44.720 --> 00:33:50.950
T minus n, and that equals
p over 1 minus p to the m.
00:33:54.240 --> 00:33:56.987
And this is how much you're
trying to get ahead-- $100
00:33:56.987 --> 00:33:58.320
in the case of my mother-in-law.
00:34:01.230 --> 00:34:03.894
So what we've
proved-- let me state
00:34:03.894 --> 00:34:05.060
what we proved as a theorem.
00:34:24.929 --> 00:34:31.830
So we proved that if
p is less than 1/2--
00:34:31.830 --> 00:34:33.889
if you're more
likely to lose a bet
00:34:33.889 --> 00:34:40.420
than win it-- then
the probability
00:34:40.420 --> 00:34:51.150
that you win m dollars before
you lose n dollars is at most p
00:34:51.150 --> 00:34:55.510
over 1 minus p to the m.
00:34:55.510 --> 00:34:58.590
That's what we just proved.
00:34:58.590 --> 00:35:00.659
And so now you can plug
in values-- for example,
00:35:00.659 --> 00:35:01.200
for roulette.
00:35:05.260 --> 00:35:11.960
p equals 9/19, which
means that p over 1
00:35:11.960 --> 00:35:19.750
minus p-- that's going
to be 9/19 over 10/19,
00:35:19.750 --> 00:35:21.010
which is just 9/10.
00:35:24.740 --> 00:35:30.010
And if m-- the amount you
want to win-- is $100,
00:35:30.010 --> 00:35:33.165
and n is $1,000-- that's
what you start with
00:35:33.165 --> 00:35:36.420
and you're willing
to lose-- well,
00:35:36.420 --> 00:35:42.640
the probability you win--
you go home happy-- W
00:35:42.640 --> 00:35:50.310
star you win $100-- is
less than or equal to 9/10
00:35:50.310 --> 00:35:53.800
raised to the m, which is 100.
00:35:53.800 --> 00:35:57.610
So it's 9/10 of 100, and that
turns out to be less than 1
00:35:57.610 --> 00:36:06.280
in 37,648, which is where
that answer came from.
00:36:06.280 --> 00:36:07.800
Now you can see why
my mother-in-law
00:36:07.800 --> 00:36:11.640
may have got lost somewhere
here now in the calculations.
00:36:11.640 --> 00:36:13.560
But this is a proof
that the chance
00:36:13.560 --> 00:36:18.850
you win $100 before you lose
$1,000 is very, very small.
00:36:18.850 --> 00:36:22.810
Now, do you see why the answer
is no better than if you came
00:36:22.810 --> 00:36:26.310
with $1 million in your pocket?
00:36:26.310 --> 00:36:30.390
Say you came with n
equals $1 million.
00:36:30.390 --> 00:36:32.390
Why is the answer not changing?
00:36:35.690 --> 00:36:37.020
Yeah.
00:36:37.020 --> 00:36:39.930
AUDIENCE: Once you
lose, say, $1,000,
00:36:39.930 --> 00:36:42.650
you're already in
a really deep hole.
00:36:42.650 --> 00:36:44.130
PROFESSOR: That's the intuition.
00:36:44.130 --> 00:36:44.671
That's right.
00:36:44.671 --> 00:36:46.560
We're going to get
to that in a minute.
00:36:46.560 --> 00:36:49.590
I want to know from
the formula, why
00:36:49.590 --> 00:36:54.280
is it no difference if I come
with $1,000 versus $1 million?
00:36:54.280 --> 00:36:55.051
Yeah.
00:36:55.051 --> 00:36:56.592
AUDIENCE: The formula
doesn't have n.
00:36:56.592 --> 00:37:00.180
PROFESSOR: Yeah, the formula
has nothing to do with n.
00:37:00.180 --> 00:37:04.130
You could come with $100
trillion in your wallet,
00:37:04.130 --> 00:37:06.710
and it doesn't
improve this bound.
00:37:06.710 --> 00:37:10.260
This bound only depends on
what you're trying to win,
00:37:10.260 --> 00:37:11.530
not on how much you came with.
00:37:11.530 --> 00:37:13.890
So no matter how
much you come with,
00:37:13.890 --> 00:37:17.250
the chance you win $100
before you lose everything
00:37:17.250 --> 00:37:20.860
is at most 1 in 37,000.
00:37:20.860 --> 00:37:22.650
Now, we can plug in
some other values
00:37:22.650 --> 00:37:27.081
just for fun--
different values of m.
00:37:32.380 --> 00:37:36.350
If you thought 1 in
37,000 was unlikely,
00:37:36.350 --> 00:37:44.380
the chance of winning
$1,000, or 1,000 bets worth
00:37:44.380 --> 00:37:50.870
before you're broke-- that's
less than 9/10 to the 1,000.
00:37:50.870 --> 00:37:57.090
That's less than 2 times
10 to the minus 46--
00:37:57.090 --> 00:37:59.760
really, really, really unlikely.
00:37:59.760 --> 00:38:06.400
Even winning $10 is not likely.
00:38:06.400 --> 00:38:07.650
Just plug in the numbers.
00:38:07.650 --> 00:38:13.790
The probability you win
$10 betting $1 at a time
00:38:13.790 --> 00:38:18.470
is less than 9/10
to the 10th power.
00:38:18.470 --> 00:38:24.150
That's less than 0.35.
00:38:24.150 --> 00:38:30.710
You can come to the casino with
$10 million, bet $1 at a time,
00:38:30.710 --> 00:38:34.390
and you quit if you just
get up 10 bets-- get up $10.
00:38:34.390 --> 00:38:38.740
The chance you get up $10 before
you lose $10 million is about 1
00:38:38.740 --> 00:38:42.100
in 3 you're twice as
likely to lose $10 million
00:38:42.100 --> 00:38:44.400
as you are to win 10.
00:38:44.400 --> 00:38:46.780
That just seems weird, right?
00:38:46.780 --> 00:38:49.250
Because it's almost a fair game.
00:38:49.250 --> 00:38:51.390
It's almost 50-50.
00:38:51.390 --> 00:38:52.820
Any questions
about the analysis?
00:38:55.860 --> 00:38:59.710
Yes, I find that shocking.
00:38:59.710 --> 00:39:02.580
Just the intuition would
seem say otherwise.
00:39:02.580 --> 00:39:03.990
So I guess there's a moral here.
00:39:03.990 --> 00:39:05.880
If you're going to
gamble, learn how
00:39:05.880 --> 00:39:09.050
to count cards in blackjack,
or some game where
00:39:09.050 --> 00:39:10.722
you can make it even.
00:39:10.722 --> 00:39:12.680
Because even in a game
where it's pretty close,
00:39:12.680 --> 00:39:14.580
you're doomed.
00:39:14.580 --> 00:39:18.710
You're just never
going to go home happy.
00:39:18.710 --> 00:39:21.900
Now, if you could
have a fair game,
00:39:21.900 --> 00:39:24.780
the world changes-- much
better circumstance.
00:39:24.780 --> 00:39:28.800
So actually, let's do the
same analysis for a fair game,
00:39:28.800 --> 00:39:31.680
because that's where our
intuition really comes from.
00:39:31.680 --> 00:39:34.550
Because we're thinking of
this game as almost fair.
00:39:34.550 --> 00:39:38.700
And in a fair game, the answer's
going to be very different.
00:39:38.700 --> 00:39:41.320
And it all goes back to the
recurrence and the roots
00:39:41.320 --> 00:39:43.490
of the characteristic equation.
00:39:43.490 --> 00:39:49.380
Because in a fair
game, p is 1/2.
00:39:53.490 --> 00:39:56.290
And then you have a double root.
00:39:56.290 --> 00:40:04.970
1 minus 1/2 over 1/2 equals 1,
and that means a double root
00:40:04.970 --> 00:40:07.570
at 1.
00:40:07.570 --> 00:40:10.390
And that changes everything.
00:40:10.390 --> 00:40:16.120
So let's go through now
and do all this analysis
00:40:16.120 --> 00:40:17.440
in the case of a fair game.
00:40:20.140 --> 00:40:22.180
And this will give us
practice with double roots
00:40:22.180 --> 00:40:23.630
and recurrences.
00:40:23.630 --> 00:40:25.939
Because as you see
now, it does happen.
00:40:30.930 --> 00:40:33.866
Let's figure out the chance
that we go home a winner.
00:40:38.490 --> 00:40:40.360
OK, so let's see.
00:40:40.360 --> 00:40:44.766
In this case, we know the roots.
00:40:44.766 --> 00:40:46.580
Can anybody tell me
what formula we're
00:40:46.580 --> 00:40:49.870
going to use for the solution?
00:40:49.870 --> 00:40:52.230
Got a double root at 1.
00:40:52.230 --> 00:40:54.170
So there's going to
be a 1 to the n here.
00:40:56.750 --> 00:40:58.590
I don't just put a
constant A in front.
00:40:58.590 --> 00:41:00.744
What do I do with a double root?
00:41:00.744 --> 00:41:01.660
AUDIENCE: [INAUDIBLE].
00:41:01.660 --> 00:41:02.260
AUDIENCE: A n.
00:41:02.260 --> 00:41:03.430
PROFESSOR: What is it?
00:41:03.430 --> 00:41:04.230
AUDIENCE: A n.
00:41:04.230 --> 00:41:06.560
PROFESSOR: A n-- not quite A n.
00:41:06.560 --> 00:41:08.200
You got an A n here.
00:41:08.200 --> 00:41:09.080
AUDIENCE: Plus B.
00:41:09.080 --> 00:41:13.280
PROFESSOR: Plus B-- that's
what you do for a double root,
00:41:13.280 --> 00:41:17.340
because you make a first
degree polynomial in n here.
00:41:17.340 --> 00:41:18.782
So we plug that in.
00:41:18.782 --> 00:41:20.240
The root's at 1,
so it's real easy.
00:41:20.240 --> 00:41:21.910
The solution's really easy now.
00:41:21.910 --> 00:41:23.340
No messy powers or anything.
00:41:23.340 --> 00:41:28.487
It's just A n plus B. And
I can figure out A and B
00:41:28.487 --> 00:41:29.695
from the boundary conditions.
00:41:36.550 --> 00:41:40.590
All right, X0 is 0.
00:41:40.590 --> 00:41:46.280
X 0 is just B, because
it's A times 0 goes away.
00:41:46.280 --> 00:41:48.120
And that means that B equals 0.
00:41:48.120 --> 00:41:50.960
This is getting really simple.
00:41:50.960 --> 00:41:56.850
1 is X T. And that's
A plus B, but B was 0.
00:41:56.850 --> 00:42:03.620
So that's A times 1
plus B. That's just A.
00:42:03.620 --> 00:42:11.600
It means A equals A n.
00:42:11.600 --> 00:42:13.270
Good, n's not 1.
00:42:13.270 --> 00:42:15.410
N's T. So it's A T plus B.
00:42:15.410 --> 00:42:17.550
This is A T here.
00:42:17.550 --> 00:42:19.280
So A T equals 1.
00:42:19.280 --> 00:42:23.780
That means A is 1 over T.
00:42:23.780 --> 00:42:32.854
All right, that means that
X n is n over T. And T
00:42:32.854 --> 00:42:33.820
is the total.
00:42:33.820 --> 00:42:36.140
The top limit is n plus
m, because you quit
00:42:36.140 --> 00:42:38.090
if you get ahead m dollars.
00:42:38.090 --> 00:42:44.030
This is just now
n over n plus m.
00:42:44.030 --> 00:42:47.880
All right, so let's
write that down.
00:42:47.880 --> 00:42:48.515
It's a theorem.
00:42:53.450 --> 00:42:58.920
If p is 1/2, i.e., you
have a fair game, then
00:42:58.920 --> 00:43:04.580
the probability
you win m dollars
00:43:04.580 --> 00:43:15.270
before you lose n dollars
is just n over n plus m.
00:43:15.270 --> 00:43:17.590
And this might fit
the intuition better.
00:43:20.700 --> 00:43:28.810
So for the mother-in-law
strategy, if m is 100,
00:43:28.810 --> 00:43:34.580
and n is 1,000, what's
the probability you
00:43:34.580 --> 00:43:36.460
win-- you go home a winner?
00:43:42.230 --> 00:43:48.904
Yeah, 1,000 over 1,000 plus 100.
00:43:48.904 --> 00:43:54.490
1,000 over 1,000 is 10 over 11.
00:43:54.490 --> 00:43:57.850
So she does go home happy most
of the time-- 10 out of 11
00:43:57.850 --> 00:44:00.410
nights-- if she's
playing a fair game.
00:44:02.970 --> 00:44:06.920
Any questions about that?
00:44:06.920 --> 00:44:13.630
So the trouble we get into
here is that the fair game
00:44:13.630 --> 00:44:16.890
results match our intuition.
00:44:16.890 --> 00:44:20.520
You know if you have 10 times
as much money in a fair game,
00:44:20.520 --> 00:44:23.660
you'd expect to go home
happy 10 out of 11 nights.
00:44:23.660 --> 00:44:24.790
That makes a lot of sense.
00:44:24.790 --> 00:44:28.660
You go home happy 10, and
then you lose the 11th.
00:44:28.660 --> 00:44:30.790
That's a 10 to 1 ratio,
which is the money
00:44:30.790 --> 00:44:33.001
you brought into the game.
00:44:33.001 --> 00:44:36.160
The trouble we get
into is, the fair game
00:44:36.160 --> 00:44:38.790
is very close to the real game.
00:44:38.790 --> 00:44:42.120
Instead of 50-50, it's 47-53.
00:44:42.120 --> 00:44:44.310
And so our intuition
says the results--
00:44:44.310 --> 00:44:46.780
the probability of going
home happy in a fair game--
00:44:46.780 --> 00:44:49.200
should be close to the
probability of going
00:44:49.200 --> 00:44:51.180
home happy in the real game.
00:44:51.180 --> 00:44:52.340
And that's not true.
00:44:52.340 --> 00:44:55.990
There's a discontinuity here
because of the double root.
00:44:55.990 --> 00:44:58.350
And the character
completely changes.
00:44:58.350 --> 00:45:01.380
So instead of being
close to 10 out of 11,
00:45:01.380 --> 00:45:04.140
you're down there
at 1 in 37,000--
00:45:04.140 --> 00:45:07.640
completely different behavior.
00:45:07.640 --> 00:45:12.580
OK, any questions?
00:45:12.580 --> 00:45:14.398
All right, so let me
give you an-- yeah.
00:45:14.398 --> 00:45:17.540
AUDIENCE: So what
happens if you make n 1,
00:45:17.540 --> 00:45:20.520
and then you do that repeatedly?
00:45:20.520 --> 00:45:23.280
PROFESSOR: Now, if
I did n equals 1,
00:45:23.280 --> 00:45:24.750
I could use that
as an upper bound,
00:45:24.750 --> 00:45:28.080
and it's not so
interesting as, say, 90%.
00:45:28.080 --> 00:45:30.890
But I would actually go
plug it back in here.
00:45:30.890 --> 00:45:33.300
So this would be n plus
1, and it would depend
00:45:33.300 --> 00:45:34.890
how much money I brought.
00:45:34.890 --> 00:45:39.640
But there is a pretty good
chance I go home a winner for m
00:45:39.640 --> 00:45:41.230
equals 1.
00:45:41.230 --> 00:45:43.940
Because I've got a pretty
good chance that I either--
00:45:43.940 --> 00:45:46.090
47% chance I win the first time.
00:45:46.090 --> 00:45:47.752
Then I go home happy.
00:45:47.752 --> 00:45:50.790
If I lost the first time, now
I've just got to win twice.
00:45:50.790 --> 00:45:52.720
And I might win twice in a row.
00:45:52.720 --> 00:45:55.330
That'll happen about
20% of the time.
00:45:55.330 --> 00:45:58.940
If I lose that, now I've
got to win three in a row.
00:45:58.940 --> 00:46:02.150
That'll happen around
10% of the time.
00:46:02.150 --> 00:46:05.540
So I've got 10 plus
20 plus almost 50.
00:46:05.540 --> 00:46:08.010
Most of the time, I'm going
to go home happy if I just
00:46:08.010 --> 00:46:10.460
have to get ahead by $1.
00:46:10.460 --> 00:46:12.460
But it doesn't take
much more than one
00:46:12.460 --> 00:46:14.660
before you're not
likely to go home happy.
00:46:14.660 --> 00:46:19.050
Getting ahead 10 is not
going to happen, very likely.
00:46:19.050 --> 00:46:21.930
Now, you want to
recurse on that?
00:46:21.930 --> 00:46:25.490
I'm pretty likely
to get ahead by one.
00:46:25.490 --> 00:46:26.990
Well, OK, get ahead by one.
00:46:26.990 --> 00:46:29.639
I'm pretty likely
to do it again.
00:46:29.639 --> 00:46:30.430
And I did it again.
00:46:30.430 --> 00:46:32.320
Now I'm pretty likely
to do it again.
00:46:32.320 --> 00:46:33.945
And there's this
thing called induction
00:46:33.945 --> 00:46:35.270
that we worried a lot about.
00:46:35.270 --> 00:46:38.130
So by induction, are we likely
to go home happy with 10?
00:46:38.130 --> 00:46:43.460
No, because every time you
don't get there, you're dead.
00:46:43.460 --> 00:46:45.720
You had a little chance of
dying and not reaching one,
00:46:45.720 --> 00:46:48.251
and a little chance of dying
and not going from one to two.
00:46:48.251 --> 00:46:50.000
And you add up all
those chances of dying,
00:46:50.000 --> 00:46:52.440
and you're toast, because
that'll be adding up
00:46:52.440 --> 00:46:56.317
to everything, pretty much.
00:46:56.317 --> 00:46:57.400
So that's a good question.
00:46:57.400 --> 00:46:59.410
If you're likely
to get up by one,
00:46:59.410 --> 00:47:01.839
why aren't you likely
to get up by 10?
00:47:01.839 --> 00:47:02.880
It doesn't work that way.
00:47:02.880 --> 00:47:05.300
That's a great question.
00:47:05.300 --> 00:47:12.130
Let me show you the phenomenon
that's going on here, as
00:47:12.130 --> 00:47:15.300
to why it works out this way.
00:47:15.300 --> 00:47:16.644
We had the math.
00:47:16.644 --> 00:47:17.810
So we looked at it that way.
00:47:17.810 --> 00:47:20.810
We notice that one case is a
double root and the other case
00:47:20.810 --> 00:47:22.200
isn't.
00:47:22.200 --> 00:47:23.640
And that exponential,
in the case
00:47:23.640 --> 00:47:25.670
where you didn't have
that second root at 1
00:47:25.670 --> 00:47:28.590
makes an enormous difference.
00:47:28.590 --> 00:47:32.800
Qualitatively, we can
draw the two cases.
00:47:32.800 --> 00:47:43.010
So in the case of an
unbiased or fair game,
00:47:43.010 --> 00:47:47.640
if we track what's
going on over time,
00:47:47.640 --> 00:47:54.265
and we start with n dollars,
sort of this is our baseline.
00:47:57.350 --> 00:48:02.190
And here's our
target-- T is n plus m.
00:48:02.190 --> 00:48:04.230
And so we quit if
we ever get here.
00:48:04.230 --> 00:48:06.530
And we quit if we
ever hit the bottom.
00:48:06.530 --> 00:48:08.920
And we've got a random walk.
00:48:08.920 --> 00:48:15.200
It's going around, just
doing this kind of stuff.
00:48:15.200 --> 00:48:19.150
And eventually, it's going to
hit one of these boundaries.
00:48:19.150 --> 00:48:23.580
And if m is small
compared to n, we're
00:48:23.580 --> 00:48:26.170
more likely to
hit this boundary.
00:48:26.170 --> 00:48:29.150
And in fact, the chance
we hit this boundary first
00:48:29.150 --> 00:48:31.120
is the ratio of these sizes.
00:48:31.120 --> 00:48:34.230
It's n over the total.
00:48:34.230 --> 00:48:38.300
It's the chance that
we hit that one first.
00:48:38.300 --> 00:48:42.300
Now in the biased case, the
picture looks different.
00:48:55.060 --> 00:49:02.342
So in the biased case--
so this is now biased.
00:49:02.342 --> 00:49:04.300
And we're going to assume
it's downward biased.
00:49:04.300 --> 00:49:05.425
You're more likely to lose.
00:49:09.960 --> 00:49:13.770
So you start at n, you've
got your boundary up here
00:49:13.770 --> 00:49:17.740
at T equals n plus m.
00:49:17.740 --> 00:49:21.190
Time is going this way.
00:49:21.190 --> 00:49:28.810
The problem is, you've got
a downward sort of baseline,
00:49:28.810 --> 00:49:33.080
because you expect to lose
a little bit each time.
00:49:33.080 --> 00:49:35.785
And so you're taking
this random walk.
00:49:38.360 --> 00:49:41.970
And you collide here.
00:49:41.970 --> 00:49:47.050
And these things are
known as the swings.
00:49:47.050 --> 00:49:48.795
This is known as the drift.
00:49:52.390 --> 00:49:55.930
And the drift downward
is 1 minus 2p.
00:49:55.930 --> 00:49:58.510
That's what you expect to
lose if you get the expected
00:49:58.510 --> 00:50:01.690
loss on each bet-- 1 minus 2p.
00:50:01.690 --> 00:50:04.800
Because you're going
to not be a fair game.
00:50:04.800 --> 00:50:06.810
This one has zero
drift up there.
00:50:06.810 --> 00:50:08.590
It stays steady.
00:50:08.590 --> 00:50:17.195
And in random walks, drift
outweighs the swings.
00:50:19.900 --> 00:50:21.420
These are the swings here.
00:50:21.420 --> 00:50:23.310
And they're random.
00:50:23.310 --> 00:50:24.980
The drift is deterministic.
00:50:24.980 --> 00:50:26.950
It's steadily going down.
00:50:26.950 --> 00:50:28.890
And so almost always
in a random walk,
00:50:28.890 --> 00:50:32.520
the drift totally
takes over the swings.
00:50:32.520 --> 00:50:34.400
The swings are small
compared to what you're
00:50:34.400 --> 00:50:37.730
losing on a steady basis.
00:50:37.730 --> 00:50:41.260
And that's why you're so
much more likely to lose when
00:50:41.260 --> 00:50:44.110
you have the drift downward.
00:50:46.670 --> 00:50:49.450
Just as an example, maybe
putting some numbers
00:50:49.450 --> 00:50:49.950
around that.
00:50:53.680 --> 00:50:55.700
The swings are the
same in both cases,
00:50:55.700 --> 00:50:59.180
So that gives you some
qualification for how big
00:50:59.180 --> 00:51:00.308
the swings tend to be.
00:51:03.660 --> 00:51:06.365
We can sort of do that with
standard deviation notation.
00:51:09.380 --> 00:51:17.510
After X bets or X steps,
the amount you've drifted,
00:51:17.510 --> 00:51:21.530
or the expected
losses, 1 minus 2p
00:51:21.530 --> 00:51:25.940
X. Maybe we should
just understand
00:51:25.940 --> 00:51:27.300
why this is the case.
00:51:27.300 --> 00:51:37.960
The expected return on a
bet is 1 with probability p,
00:51:37.960 --> 00:51:41.650
and minus 1 with
probability 1 minus p.
00:51:41.650 --> 00:51:46.046
And so that is-- did
I get that right?
00:51:46.046 --> 00:51:49.730
I think that's right.
00:51:49.730 --> 00:51:52.200
Oh, expected loss--
[INAUDIBLE] drifts down.
00:51:52.200 --> 00:51:54.569
Instead of expected
return, let's do the loss,
00:51:54.569 --> 00:51:55.610
because that's the drift.
00:51:55.610 --> 00:51:57.270
It's a downward thing.
00:51:57.270 --> 00:52:01.910
So the expected loss-- now
you lose $1 with 1 minus p.
00:52:04.610 --> 00:52:07.670
And you gain $1, which
is negative loss,
00:52:07.670 --> 00:52:10.050
with probability p.
00:52:10.050 --> 00:52:14.570
And so you get 1 minus
p minus p is 1 minus 2p.
00:52:14.570 --> 00:52:16.370
So that's your expected loss.
00:52:16.370 --> 00:52:21.300
Your expected winnings
are the negative of that.
00:52:21.300 --> 00:52:25.700
So after x steps, you
expect to lose-- well,
00:52:25.700 --> 00:52:27.700
I just add up the
linearity of expectation.
00:52:27.700 --> 00:52:30.950
You expect to lose
this much x times.
00:52:30.950 --> 00:52:32.480
So that's your expected drift.
00:52:32.480 --> 00:52:36.440
You're expected
to lose that much.
00:52:36.440 --> 00:52:38.950
Now, the swing--
and we won't prove
00:52:38.950 --> 00:52:44.670
this-- the swing is expected
to be square root of x
00:52:44.670 --> 00:52:45.480
times a constant.
00:52:45.480 --> 00:52:47.670
So I've used the
theta notation here.
00:52:47.670 --> 00:52:50.230
And the constant is small.
00:52:50.230 --> 00:52:54.090
If I take x consecutive
bets for $1,
00:52:54.090 --> 00:52:57.610
I'm very likely to be
about square root of x
00:52:57.610 --> 00:53:01.510
off of the expected drift.
00:53:01.510 --> 00:53:05.040
And you can see that
this is square root.
00:53:05.040 --> 00:53:06.550
That is linear.
00:53:06.550 --> 00:53:10.250
So this totally dominates that.
00:53:10.250 --> 00:53:14.110
So the swings are generally
not enough to save you.
00:53:14.110 --> 00:53:17.300
And so you're just going to
cruise downward and crash,
00:53:17.300 --> 00:53:18.136
almost surely.
00:53:21.330 --> 00:53:25.160
OK, any questions about that?
00:53:31.360 --> 00:53:34.430
All right, so we figured out
the probability of winning m
00:53:34.430 --> 00:53:36.420
dollars before going broke.
00:53:36.420 --> 00:53:40.230
That's done with.
00:53:40.230 --> 00:53:43.060
Now, this means it's
logical to conclude
00:53:43.060 --> 00:53:47.310
you're likely go home
broke in an unfair game.
00:53:47.310 --> 00:53:50.820
Actually, before we do
that, there's one other case
00:53:50.820 --> 00:53:52.760
we've got to rule out.
00:53:52.760 --> 00:53:56.512
We've proved you're likely
not to go home a winner.
00:53:56.512 --> 00:53:58.720
Does that necessarily mean
you're likely to go broke?
00:53:58.720 --> 00:54:00.845
I've been saying that, but
there's some other thing
00:54:00.845 --> 00:54:03.240
we should check.
00:54:03.240 --> 00:54:07.484
What's one way you
might not go home broke?
00:54:07.484 --> 00:54:08.400
AUDIENCE: [INAUDIBLE].
00:54:08.400 --> 00:54:09.385
PROFESSOR: What is it?
00:54:09.385 --> 00:54:09.780
AUDIENCE: You don't go home.
00:54:09.780 --> 00:54:11.220
PROFESSOR: You don't go home.
00:54:11.220 --> 00:54:13.520
And why would you not go home?
00:54:13.520 --> 00:54:14.020
Yeah?
00:54:14.020 --> 00:54:15.395
AUDIENCE: You're
playing forever.
00:54:15.395 --> 00:54:16.990
PROFESSOR: You're
playing forever-- we
00:54:16.990 --> 00:54:19.840
didn't rule out that case--
you're playing forever.
00:54:19.840 --> 00:54:22.790
But it turns out, if you
did the same analysis,
00:54:22.790 --> 00:54:25.200
you can analyze the probability
of going home broke.
00:54:25.200 --> 00:54:27.366
And when you add it to the
probability of going home
00:54:27.366 --> 00:54:30.320
a winner, it adds to 1,
which means the probability
00:54:30.320 --> 00:54:33.480
playing forever is 0.
00:54:33.480 --> 00:54:35.757
Now, there are sample points
where you play forever.
00:54:35.757 --> 00:54:37.590
But when you add up all
those sample points,
00:54:37.590 --> 00:54:41.210
if their probability
is 0, we ignore them.
00:54:41.210 --> 00:54:42.950
And we say it can't happen.
00:54:42.950 --> 00:54:46.379
Now, we're bordering
on philosophy here,
00:54:46.379 --> 00:54:47.920
because there is a
sample point here.
00:54:47.920 --> 00:54:50.640
You could win, lose, win,
lose, win, lose forever.
00:54:50.640 --> 00:54:54.305
But because you add them
all up at 0, measure theory
00:54:54.305 --> 00:54:56.430
and some math we're not
going to get into tells you
00:54:56.430 --> 00:54:58.010
it doesn't happen.
00:54:58.010 --> 00:55:00.430
It's probability 1 you're
a winner or a loser.
00:55:07.430 --> 00:55:09.760
All right, so I'm
not going to prove
00:55:09.760 --> 00:55:13.350
that the probability
you play forever is 0.
00:55:24.250 --> 00:55:27.660
But let's look at
how long you play.
00:55:27.660 --> 00:55:32.230
How long does it take you to
go home one way or another-- go
00:55:32.230 --> 00:55:33.750
broke?
00:55:33.750 --> 00:55:36.440
And to do this, we're going
to set up another recurrence.
00:55:39.440 --> 00:55:42.160
So we know eventually
we hit a boundary.
00:55:42.160 --> 00:55:44.160
I want to know how
many bets does it
00:55:44.160 --> 00:55:48.100
take to hit the boundary?
00:55:48.100 --> 00:55:50.530
How long do we get to play
before we go home unhappy?
00:55:53.270 --> 00:55:59.150
So S will be the number of
steps until we hit a boundary.
00:56:04.320 --> 00:56:06.820
And I want to know the
expected number-- I'll call it
00:56:06.820 --> 00:56:10.464
E sub n here-- is the
expected value of S given
00:56:10.464 --> 00:56:11.630
that I start with n dollars.
00:56:14.884 --> 00:56:16.300
I mean, the reason
you could think
00:56:16.300 --> 00:56:18.550
about this is, we know
we're going to go home
00:56:18.550 --> 00:56:20.410
broke-- pretty likely.
00:56:20.410 --> 00:56:22.810
Do we at least have some
fun in the meantime?
00:56:22.810 --> 00:56:25.640
Do we get a lot gambling in
and free drinks, or whatever,
00:56:25.640 --> 00:56:28.700
before we're killed here?
00:56:31.590 --> 00:56:33.210
Now, this also has a recurrence.
00:56:33.210 --> 00:56:35.030
And I'm going to show
you what it is, then
00:56:35.030 --> 00:56:37.940
prove that that's correct.
00:56:37.940 --> 00:56:42.610
So I claim that
the expected number
00:56:42.610 --> 00:56:44.600
of steps given we
start with n dollars
00:56:44.600 --> 00:56:51.500
is 0 if we start with no money,
because we are already broke.
00:56:51.500 --> 00:56:57.370
It's 0 if we start
with T dollars,
00:56:57.370 --> 00:56:59.110
because then we
just go home happy.
00:56:59.110 --> 00:57:03.290
There's no bets, because we've
already hit the upper boundary.
00:57:03.290 --> 00:57:09.490
And the interesting case will be
it's 1 plus p times E n minus 1
00:57:09.490 --> 00:57:17.980
plus 1 minus p-- oops, n plus
1-- 1 minus p E n minus 1,
00:57:17.980 --> 00:57:21.423
if we start with
between 0 and T dollars.
00:57:26.980 --> 00:57:30.650
OK, so let's prove that.
00:57:33.400 --> 00:57:38.660
Actually, the proof is exactly
the same as the last one.
00:57:38.660 --> 00:57:40.194
So I don't think
I need to do it.
00:57:44.300 --> 00:57:48.370
The proof is pretty simple,
because we look at two cases.
00:57:48.370 --> 00:57:52.150
You win the first bet--
happens with probability p.
00:57:52.150 --> 00:57:55.250
And then you're starting
with n plus $1 over again.
00:57:55.250 --> 00:57:58.050
Or you lose the
first bet-- happens
00:57:58.050 --> 00:58:00.360
with probability 1 minus p.
00:58:00.360 --> 00:58:03.890
And you're starting
over with n minus $1
00:58:03.890 --> 00:58:05.705
now-- same as last time.
00:58:05.705 --> 00:58:07.080
In fact, this
whole recurrence is
00:58:07.080 --> 00:58:11.690
identical to last time
except for one thing.
00:58:11.690 --> 00:58:13.914
What's the one thing
that's different now?
00:58:13.914 --> 00:58:14.870
AUDIENCE: [INAUDIBLE].
00:58:14.870 --> 00:58:15.994
PROFESSOR: What is it?
00:58:15.994 --> 00:58:18.780
AUDIENCE: You have [INAUDIBLE].
00:58:18.780 --> 00:58:20.738
PROFESSOR: You have--
00:58:20.738 --> 00:58:23.208
AUDIENCE: So it's not
[INAUDIBLE] any more.
00:58:23.208 --> 00:58:24.690
PROFESSOR: That's different.
00:58:24.690 --> 00:58:26.920
There's another difference.
00:58:26.920 --> 00:58:30.100
That's one difference that's
going to make it inhomogeneous.
00:58:30.100 --> 00:58:31.409
That's sort of a pain.
00:58:31.409 --> 00:58:33.200
What's the other
difference from last time?
00:58:33.200 --> 00:58:35.025
This part's the same otherwise.
00:58:35.025 --> 00:58:35.900
AUDIENCE: Boundaries.
00:58:35.900 --> 00:58:36.400
PROFESSOR: What is it?
00:58:36.400 --> 00:58:37.650
AUDIENCE: Boundary conditions.
00:58:37.650 --> 00:58:40.120
PROFESSOR: Boundary conditions--
that was a 1 before.
00:58:40.120 --> 00:58:42.290
Now it's a 0.
00:58:42.290 --> 00:58:46.420
OK, so a little change
here, and I added a 1 here.
00:58:46.420 --> 00:58:50.041
But that's going to make it
a pretty different answer.
00:58:50.041 --> 00:58:51.540
So let's see what
the recurrence is.
00:58:51.540 --> 00:58:57.310
I'll rearrange terms here
to put it into recurrence.
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
00:59:07.650 --> 00:59:10.710
equals minus 1, not 0.
00:59:10.710 --> 00:59:15.570
And the boundary conditions
are E 0 is 0 and E T is 0.
00:59:18.210 --> 00:59:21.550
OK, what's the
first thing you do
00:59:21.550 --> 00:59:26.020
when you have an inhomogeneous
linear recurrence?
00:59:26.020 --> 00:59:28.920
Solve the homogeneous one.
00:59:28.920 --> 00:59:32.050
And the answer there-- well,
it's the same as before.
00:59:32.050 --> 00:59:33.990
This is the part we analyzed.
00:59:33.990 --> 00:59:36.650
And we'll do it for
the case when p is not
00:59:36.650 --> 00:59:39.770
1/2-- so the unfair game.
00:59:39.770 --> 00:59:47.270
So the homogeneous
solution is E n
00:59:47.270 --> 00:59:51.870
just from before-- same
thing-- 1 minus p over p
00:59:51.870 --> 00:59:55.720
to the n plus B. And this
is the case with two roots.
00:59:55.720 --> 00:59:56.842
p does not equal 1/2.
01:00:00.620 --> 01:00:07.634
What's the next thing you do
for inhomogeneous recurrence?
01:00:07.634 --> 01:00:11.620
Are we plugging in
boundary conditions yet?
01:00:11.620 --> 01:00:12.120
No.
01:00:12.120 --> 01:00:14.840
So what do I do next?
01:00:14.840 --> 01:00:15.860
Particular solution.
01:00:21.160 --> 01:00:25.660
And what's my first guess?
01:00:25.660 --> 01:00:31.780
We have the recurrence
like this here.
01:00:34.590 --> 01:00:37.960
What do I guess for E n?
01:00:37.960 --> 01:00:42.610
I'm trying to guess something
that looks like that.
01:00:42.610 --> 01:00:45.380
So what do I guess?
01:00:45.380 --> 01:00:47.740
Constant, yeah.
01:00:47.740 --> 01:00:48.420
That's a scalar.
01:00:48.420 --> 01:00:51.120
I just guess a constant.
01:00:51.120 --> 01:00:57.420
And if I plug a constant a
into here, it's going to fail.
01:00:57.420 --> 01:01:01.730
Because I'll just
pull the a out.
01:01:01.730 --> 01:01:05.740
I'll get p minus 1
plus 1 minus p is 0,
01:01:05.740 --> 01:01:07.640
and 0 doesn't equal minus 1.
01:01:07.640 --> 01:01:08.220
So it fails.
01:01:12.240 --> 01:01:13.290
So I guess again.
01:01:13.290 --> 01:01:16.380
What do I guess next time?
01:01:16.380 --> 01:01:19.990
a n plus b.
01:01:19.990 --> 01:01:22.580
All right, and I
don't think I'll
01:01:22.580 --> 01:01:28.430
drag you through all the
algebra for that, but it works.
01:01:28.430 --> 01:01:35.310
And when you do it, you
find that a is minus 1
01:01:35.310 --> 01:01:38.100
over 2p minus 1.
01:01:38.100 --> 01:01:39.470
And b could be anything.
01:01:39.470 --> 01:01:44.080
So let me just rewrite
this as 1 over 1 minus 2p.
01:01:44.080 --> 01:01:46.170
And b can be anything, so
we'll set b equal to 0.
01:01:49.490 --> 01:01:52.910
So we've got our
particular solution.
01:01:52.910 --> 01:01:54.520
It's not hard to
go compute that.
01:01:54.520 --> 01:01:57.402
You just plug it
back in and solve.
01:02:04.642 --> 01:02:06.850
Now we add them together to
get the general solution.
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.
01:02:25.440 --> 01:02:29.270
And now what do we do to finish?
01:02:29.270 --> 01:02:33.700
I've got my general
solution here
01:02:33.700 --> 01:02:37.537
by adding up the homogeneous
and the particular solution.
01:02:37.537 --> 01:02:38.870
Plug in the boundary conditions.
01:02:45.680 --> 01:02:49.789
All right, I'm not going to drag
you through solving this case,
01:02:49.789 --> 01:02:51.330
but I'm going to
show you the answer.
01:02:55.430 --> 01:03:03.480
E n equals n over 1 minus 2p
minus T, the upper boundary,
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
01:03:13.400 --> 01:03:17.930
minus p over p to the T minus 1.
01:03:17.930 --> 01:03:20.670
So actually, this looks a little
familiar from the last time
01:03:20.670 --> 01:03:23.742
when we did this recurrence,
figuring out the probability we
01:03:23.742 --> 01:03:24.450
go home a winner.
01:03:24.450 --> 01:03:27.630
Here this is the
expected number of steps
01:03:27.630 --> 01:03:30.900
to hit a boundary, to go home.
01:03:30.900 --> 01:03:33.130
If we plug in the values,
it's a little hairy,
01:03:33.130 --> 01:03:35.850
but you can compute it.
01:03:35.850 --> 01:03:42.860
So for example, if m
is 100, n is 1,000,
01:03:42.860 --> 01:03:47.160
T would be 1,100 in that case.
01:03:47.160 --> 01:03:52.210
p is 9/19 playing roulette.
01:03:52.210 --> 01:03:59.580
Then the expected number of
bets before you have to go home
01:03:59.580 --> 01:04:13.480
is 1,900 from this part,
minus 0.56 from that part.
01:04:13.480 --> 01:04:17.570
So actually 19,000, sorry.
01:04:17.570 --> 01:04:24.580
So it's very close to 19,000
bets you've got to make.
01:04:24.580 --> 01:04:28.230
So it takes a long
time to lose $1,000.
01:04:28.230 --> 01:04:32.410
And it sort of comes
very close to the answer
01:04:32.410 --> 01:04:34.860
you would have guessed
without thinking and solving
01:04:34.860 --> 01:04:36.120
the recurrence.
01:04:36.120 --> 01:04:41.350
If you expect to lose
1 minus 2p every bet,
01:04:41.350 --> 01:04:44.160
and you want to know how long
the expected time to lose
01:04:44.160 --> 01:04:46.890
n dollars, you might
well have said,
01:04:46.890 --> 01:04:51.210
I think it's going to be n over
the amount I lose every time.
01:04:51.210 --> 01:04:54.460
That would be
wrong, technically,
01:04:54.460 --> 01:04:57.400
because you'd have left
off this nasty thing.
01:04:57.400 --> 01:05:00.780
But this nasty thing doesn't
make much of a real difference,
01:05:00.780 --> 01:05:04.245
because it goes to 0 really
fast for any numbers like 100
01:05:04.245 --> 01:05:06.570
and 1,000-- makes no
difference at all.
01:05:06.570 --> 01:05:08.380
So the intuition in
that case comes out
01:05:08.380 --> 01:05:11.430
to be pretty close, even
though technically, it's
01:05:11.430 --> 01:05:15.540
not exactly right.
01:05:15.540 --> 01:05:21.460
Now, to see why this goes to
0, if T equals n plus m here--
01:05:21.460 --> 01:05:25.350
this is n plus m--
and your upper limits,
01:05:25.350 --> 01:05:31.040
say m goes to infinity--
it's 100 in this case-- then
01:05:31.040 --> 01:05:34.770
that just zooms to 0, and
you're only left with that.
01:05:34.770 --> 01:05:39.510
Which means that we can use
asymptotic notation here
01:05:39.510 --> 01:05:42.782
to sort of characterize the
expected number of bets.
01:05:48.020 --> 01:05:51.000
And it's totally
dominated by the drift.
01:05:51.000 --> 01:05:58.120
So as m goes to infinity, the
expected time to live here
01:05:58.120 --> 01:06:02.060
is tilde n over 1 minus 2p.
01:06:02.060 --> 01:06:07.330
If you've got n dollars,
losing 1 minus 2p every time,
01:06:07.330 --> 01:06:11.650
then you last for n
over 1 minus 2p steps.
01:06:11.650 --> 01:06:17.180
OK, now, actually,
what situation in words
01:06:17.180 --> 01:06:19.900
does m going to infinity mean?
01:06:19.900 --> 01:06:23.200
Say I set m to be infinity?
01:06:23.200 --> 01:06:26.810
What is that kind of
game if m is infinity?
01:06:26.810 --> 01:06:28.120
How long am I playing now?
01:06:28.120 --> 01:06:28.729
Yeah.
01:06:28.729 --> 01:06:30.520
AUDIENCE: Now you're
playing for as long as
01:06:30.520 --> 01:06:32.760
it takes you to lose
all of your money.
01:06:32.760 --> 01:06:36.200
PROFESSOR: Yes, because there is
no stopping condition up here--
01:06:36.200 --> 01:06:37.380
going home happy.
01:06:37.380 --> 01:06:43.060
I'm going to play forever
or until I lose everything.
01:06:43.060 --> 01:06:47.460
And this says how long
you expect to play.
01:06:47.460 --> 01:06:51.220
It's a little less
than n over 1 minus 2p.
01:06:51.220 --> 01:06:53.900
So if you play
until you go broke,
01:06:53.900 --> 01:06:55.510
that's how long
you expect to play.
01:06:59.950 --> 01:07:03.200
So that sort of makes
sense in that scenario.
01:07:03.200 --> 01:07:06.570
That's not one where it
surprises you by intuition.
01:07:06.570 --> 01:07:08.904
It is interesting to consider
the case of a fair game.
01:07:08.904 --> 01:07:10.820
Because there's something
that's non-intuitive
01:07:10.820 --> 01:07:12.570
that happens there.
01:07:12.570 --> 01:07:14.290
So in a fair game, p is 1/2.
01:07:17.350 --> 01:07:24.280
Now, if I plug in 1/2
here, well, I divide by 0.
01:07:24.280 --> 01:07:27.730
I expect to play forever.
01:07:27.730 --> 01:07:29.500
That's not a good way
to do the analysis,
01:07:29.500 --> 01:07:30.950
that you get to a divide by 0.
01:07:30.950 --> 01:07:33.580
Let's actually go
back and look at this
01:07:33.580 --> 01:07:35.100
for the case when p is 1/2.
01:07:37.769 --> 01:07:39.310
And see what happens
in a fair game--
01:07:39.310 --> 01:07:43.250
how long you expect to
play in a fair game.
01:07:43.250 --> 01:07:48.500
Then the homogeneous
solution is the simple case.
01:07:48.500 --> 01:07:54.270
E is A n plus B. You
have a double root at 1,
01:07:54.270 --> 01:07:57.340
which we don't have to
worry about 1 to the n.
01:07:57.340 --> 01:08:03.550
When you do your
particular solution,
01:08:03.550 --> 01:08:08.620
you'll try a single
scalar, and it fails.
01:08:08.620 --> 01:08:11.730
I'll use lowercase a-- fails.
01:08:11.730 --> 01:08:18.460
You will then try a degree one
polynomial, and that will fail.
01:08:18.460 --> 01:08:21.010
What are you going to try next?
01:08:21.010 --> 01:08:28.210
Second-degree polynomial,
and that will work.
01:08:28.210 --> 01:08:34.100
OK, and the answer
you get when you
01:08:34.100 --> 01:08:41.960
do that is that-- I'll
put the answer here.
01:08:41.960 --> 01:08:47.450
It turns out that a is minus
1 and b and c can be 0.
01:08:47.450 --> 01:08:49.470
So it's just going
to be minus n squared
01:08:49.470 --> 01:08:51.660
for the particular solution.
01:08:51.660 --> 01:08:59.931
That means your general solution
is A n plus B minus n squared.
01:08:59.931 --> 01:09:01.389
Now you do your
boundary condition.
01:09:06.660 --> 01:09:10.180
You have E 0 is 0.
01:09:10.180 --> 01:09:11.359
Plug in 0 for n.
01:09:11.359 --> 01:09:13.720
That's equal to B. So B is 0.
01:09:13.720 --> 01:09:15.620
That's nice.
01:09:15.620 --> 01:09:19.180
E T is 0.
01:09:19.180 --> 01:09:25.029
And I plug in T here, I get
AT, B is 0 minus T squared.
01:09:25.029 --> 01:09:28.160
So I solve for A here.
01:09:28.160 --> 01:09:35.550
That means that A equals T. AT
squared minus T squared is 0.
01:09:35.550 --> 01:09:38.439
A has to be T.
01:09:38.439 --> 01:09:49.960
So that means that E n
is Tn minus n squared.
01:09:49.960 --> 01:09:52.090
Now, T is the upper bound.
01:09:52.090 --> 01:09:53.140
It's just n plus m.
01:09:56.800 --> 01:10:00.880
n plus m times n
minus n squared--
01:10:00.880 --> 01:10:01.925
this gets really simple.
01:10:01.925 --> 01:10:03.190
The m squared cancels.
01:10:03.190 --> 01:10:06.470
I just get n out.
01:10:06.470 --> 01:10:08.820
That says if you're playing
a fair game, until you
01:10:08.820 --> 01:10:14.400
win m or lose n, you expect
to play for nm steps, which
01:10:14.400 --> 01:10:15.680
is really nice.
01:10:15.680 --> 01:10:21.100
This is p is 1/2-- very clean.
01:10:21.100 --> 01:10:25.940
Now, if you let m
equal to infinity,
01:10:25.940 --> 01:10:29.680
you're going to expect
to play forever.
01:10:29.680 --> 01:10:33.680
So with a fair game, if you
play until you're broke,
01:10:33.680 --> 01:10:37.220
the expected number
of bets is infinite.
01:10:37.220 --> 01:10:39.510
That's nice.
01:10:39.510 --> 01:10:41.990
You can play forever
is the expectation.
01:10:46.160 --> 01:10:50.430
Now, here's the weird thing.
01:10:50.430 --> 01:10:53.420
If you expect to play
forever, does that
01:10:53.420 --> 01:10:55.725
mean you're not likely
to go home broke?
01:10:58.580 --> 01:11:02.410
You expect to play forever.
01:11:02.410 --> 01:11:05.976
And as long as you're playing,
you're not going home broke.
01:11:05.976 --> 01:11:07.850
Now, there's some chance
of going home broke,
01:11:07.850 --> 01:11:10.330
because you might just lose
every bet-- not likely.
01:11:15.930 --> 01:11:18.580
Here's the weird
thing-- the probability
01:11:18.580 --> 01:11:24.460
you go home broke if you
play until you go broke is 1.
01:11:24.460 --> 01:11:26.970
You will go home broke.
01:11:26.970 --> 01:11:30.140
It's just that it
takes you expected
01:11:30.140 --> 01:11:32.930
infinite amount
of time to do it--
01:11:32.930 --> 01:11:36.350
sort of one of these weird
things in a fair game.
01:11:36.350 --> 01:11:40.330
So here we proved the
expected number bets is nm.
01:11:40.330 --> 01:11:44.420
If m is infinite, that becomes
an infinite number of bets.
01:11:44.420 --> 01:11:48.095
One more theorem here-- this
one's a little surprising.
01:11:52.172 --> 01:11:54.130
This theorem is called
Quit While You're Ahead.
01:12:06.570 --> 01:12:18.700
If you start with n dollars,
and it's a fair game,
01:12:18.700 --> 01:12:33.700
and you play until
you go broke, then
01:12:33.700 --> 01:12:38.540
the probability that
you do go broke,
01:12:38.540 --> 01:12:41.730
as opposed to playing
forever, is 1.
01:12:41.730 --> 01:12:43.430
It's a certainty.
01:12:43.430 --> 01:12:47.140
You'll go broke, even
though you expect it to take
01:12:47.140 --> 01:12:49.570
an infinite amount of time.
01:12:49.570 --> 01:12:51.074
All right, so let's prove that.
01:13:06.561 --> 01:13:07.977
OK, the proof is
by contradiction.
01:13:15.330 --> 01:13:18.600
Assume it's not true.
01:13:18.600 --> 01:13:23.220
And that means that
you're assuming
01:13:23.220 --> 01:13:26.080
that there exists
some number of dollars
01:13:26.080 --> 01:13:32.880
that you can start with, and
some epsilon bigger than 0,
01:13:32.880 --> 01:13:41.190
such that the probability
that you lose the n
01:13:41.190 --> 01:13:43.900
dollars-- in which
case you're going home
01:13:43.900 --> 01:13:56.310
broke-- let me write the
probability you go broke--
01:13:56.310 --> 01:14:00.009
is at most 1 minus epsilon.
01:14:00.009 --> 01:14:01.800
In other words, if the
theorem is not true,
01:14:01.800 --> 01:14:04.860
there's some amount of money
you can start with such
01:14:04.860 --> 01:14:09.000
that the chance you go broke
is less than 1-- less than 1
01:14:09.000 --> 01:14:12.150
minus epsilon.
01:14:12.150 --> 01:14:17.590
OK, now that means that for
all m, where you might possibly
01:14:17.590 --> 01:14:20.680
stop but you're not
going to, the probability
01:14:20.680 --> 01:14:31.365
you lose n before you win m
is at most 1 minus epsilon.
01:14:35.700 --> 01:14:37.240
Because we're saying
the probability
01:14:37.240 --> 01:14:39.360
you lose n no matter
what is at most that.
01:14:39.360 --> 01:14:41.820
So it's certainly less
than 1 minus epsilon
01:14:41.820 --> 01:14:46.670
that you lose n before
you win m dollars.
01:14:46.670 --> 01:14:48.880
And we know what
that probability is.
01:14:48.880 --> 01:14:51.770
This probability is
just m over n plus m.
01:14:51.770 --> 01:14:54.700
We proved that earlier.
01:14:54.700 --> 01:14:59.360
So that has to be less than
1 minus epsilon for all m.
01:14:59.360 --> 01:15:05.150
And now I just multiply
through for all m.
01:15:05.150 --> 01:15:07.570
That means that m is
less than or equal to 1
01:15:07.570 --> 01:15:09.616
minus epsilon n plus m.
01:15:12.930 --> 01:15:17.092
And then we'll solve that.
01:15:25.960 --> 01:15:29.630
OK, so just multiply this out.
01:15:29.630 --> 01:15:39.260
So for all m less than or
equal to n plus m minus epsilon
01:15:39.260 --> 01:15:45.360
n minus epsilon m, and now
pull the m terms out here,
01:15:45.360 --> 01:15:49.640
I get for all m,
epsilon m is less than
01:15:49.640 --> 01:15:53.020
or equal to 1 minus epsilon n.
01:15:53.020 --> 01:15:55.840
That means for all
m, m is smaller
01:15:55.840 --> 01:16:01.170
than 1 minus epsilon
over epsilon times n.
01:16:01.170 --> 01:16:03.060
And that can't be true.
01:16:03.060 --> 01:16:06.270
It's not true the for all
m, this is less than that,
01:16:06.270 --> 01:16:09.030
because these are fixed values.
01:16:09.030 --> 01:16:11.790
That's a contradiction.
01:16:11.790 --> 01:16:15.320
All right, so we
proved that if you
01:16:15.320 --> 01:16:17.530
keep playing until
you're broke, you will go
01:16:17.530 --> 01:16:20.030
broke with probability 1.
01:16:20.030 --> 01:16:25.060
So even if you're playing a fair
game, quit while you're ahead.
01:16:25.060 --> 01:16:28.970
Because if you don't,
you're going to go broke.
01:16:28.970 --> 01:16:31.500
The swings will eventually
catch up with you.
01:16:31.500 --> 01:16:35.159
So if we draw the graph here,
we'll see why that's true.
01:16:51.810 --> 01:16:55.660
All right, if I have
time going this way,
01:16:55.660 --> 01:17:01.750
and I start with n dollars,
my baseline is here.
01:17:01.750 --> 01:17:03.884
The drift is 0.
01:17:03.884 --> 01:17:04.925
I'm going to have swings.
01:17:07.480 --> 01:17:09.510
I might have some
really big, high swings,
01:17:09.510 --> 01:17:12.700
but it doesn't matter,
because eventually I'm
01:17:12.700 --> 01:17:16.030
going to get a really bad swing,
and I'm going to go broke.
01:17:19.450 --> 01:17:21.190
Now, if you ever
play a game where
01:17:21.190 --> 01:17:25.960
you're likely to be winning each
time, and the drift goes up,
01:17:25.960 --> 01:17:27.635
that's a good game
to play, obviously.
01:17:27.635 --> 01:17:29.529
It just keeps getting better.
01:17:29.529 --> 01:17:31.070
And that's a whole
math change there.
01:17:34.410 --> 01:17:36.380
So that's it.
01:17:36.380 --> 01:17:39.030
Remember, we have the ice
cream study session Monday.
01:17:39.030 --> 01:17:41.790
So come to that if you'd like.
01:17:41.790 --> 01:17:45.470
And definitely come to
the final on Tuesday.
01:17:45.470 --> 01:17:47.440
And thanks for your
hard work, and being
01:17:47.440 --> 01:17:49.170
such a great class this year.
01:17:49.170 --> 01:17:52.820
[APPLAUSE]