WEBVTT
00:00:00.499 --> 00:00:02.820
The following content is
provided under a Creative
00:00:02.820 --> 00:00:04.340
Commons license.
00:00:04.340 --> 00:00:06.670
Your support will help
MIT OpenCourseWare
00:00:06.670 --> 00:00:11.040
continue to offer high quality
educational resources for free.
00:00:11.040 --> 00:00:13.650
To make a donation or
view additional materials
00:00:13.650 --> 00:00:17.537
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.537 --> 00:00:18.162
at ocw.mit.edu.
00:00:23.160 --> 00:00:25.370
PROFESSOR: Just a reminder,
drop day is tomorrow.
00:00:25.370 --> 00:00:27.415
So if you were thinking
about dropping the course
00:00:27.415 --> 00:00:29.960
or in danger of a bad
grade or something,
00:00:29.960 --> 00:00:34.420
tomorrow's the last
chance to bail out.
00:00:34.420 --> 00:00:36.660
Last time we began our
discussion on probability
00:00:36.660 --> 00:00:40.440
with the Monty Hall game--
the Monty Hall problem.
00:00:40.440 --> 00:00:43.280
And as part of the analysis,
we made assumptions
00:00:43.280 --> 00:00:48.580
of the form that given that
Carol placed the prize in box
00:00:48.580 --> 00:00:52.440
1, the probability that
the contestant chooses
00:00:52.440 --> 00:00:55.280
box 1 is 1/3.
00:00:55.280 --> 00:00:57.620
Now, this is an
example of something
00:00:57.620 --> 00:01:00.589
that's called a
conditional probability.
00:01:00.589 --> 00:01:02.380
And that's what we're
going to study today.
00:01:13.930 --> 00:01:16.170
Now, in general,
you have something
00:01:16.170 --> 00:01:19.950
like the conditional
probability that an event, A,
00:01:19.950 --> 00:01:24.210
happens given that some
other event, B, has already
00:01:24.210 --> 00:01:25.770
taken place.
00:01:25.770 --> 00:01:38.650
And you write that down as
a probability of A given B.
00:01:38.650 --> 00:01:39.945
And both A and B are events.
00:01:44.190 --> 00:01:46.665
Now, the example
from Monty Hall--
00:01:46.665 --> 00:01:48.870
and actually, we had
several-- but you
00:01:48.870 --> 00:01:56.230
might have B being
the event that Carol
00:01:56.230 --> 00:01:59.670
places the prize in box 1.
00:02:03.820 --> 00:02:17.300
And A might be the event that
the contestant chooses box 1.
00:02:17.300 --> 00:02:20.200
And we assumed for
the Monty Hall game
00:02:20.200 --> 00:02:29.200
that the probability of
A given B in this case
00:02:29.200 --> 00:02:33.110
was 1/3 third because
the contestant didn't
00:02:33.110 --> 00:02:35.220
know where the prize was.
00:02:37.960 --> 00:02:40.320
Now in general, there's
a very simple formula
00:02:40.320 --> 00:02:44.974
to compute the probability
of A given B. In fact, we'll
00:02:44.974 --> 00:02:46.015
treat it as a definition.
00:02:48.920 --> 00:02:55.700
Assuming the probability of B
is non-zero than the probability
00:02:55.700 --> 00:03:03.440
of A given B is just the
probability of A and B
00:03:03.440 --> 00:03:07.860
happening, both happening,
divided by the probability
00:03:07.860 --> 00:03:10.700
of B happening.
00:03:10.700 --> 00:03:15.620
And you can see why this makes
sense when the picture-- say
00:03:15.620 --> 00:03:18.560
this is our sample space.
00:03:18.560 --> 00:03:22.370
And let this be the
event, A, and this
00:03:22.370 --> 00:03:28.440
be the event, B. Now we're
conditioning on the fact
00:03:28.440 --> 00:03:31.110
that B happened.
00:03:31.110 --> 00:03:33.360
Now once we've
conditioned on that,
00:03:33.360 --> 00:03:37.320
all this stuff outside of
B is no longer possible.
00:03:37.320 --> 00:03:42.930
All those outcomes are no longer
in the space of consideration.
00:03:42.930 --> 00:03:45.850
The only outcomes left
are in B. So in some sense
00:03:45.850 --> 00:03:49.050
we've shrunk the
sample space to be B.
00:03:49.050 --> 00:03:52.660
And all we care about is the
probability that A happens
00:03:52.660 --> 00:03:55.410
inside this new sample space.
00:03:55.410 --> 00:03:58.950
And that is, we're asking the
probability 1 of these outcomes
00:03:58.950 --> 00:04:03.040
happens given that this
is the sample space.
00:04:03.040 --> 00:04:07.160
Well, this is just A
intersect B because you still
00:04:07.160 --> 00:04:09.650
have to have A happen, but
now you're inside of B.
00:04:09.650 --> 00:04:13.530
And then we divide
by probability of B.
00:04:13.530 --> 00:04:17.150
So we normalize this
to be probability one.
00:04:17.150 --> 00:04:17.649
OK.
00:04:17.649 --> 00:04:19.490
Because we're
saying B happened--
00:04:19.490 --> 00:04:20.970
we're conditioning on that.
00:04:20.970 --> 00:04:24.680
Therefore, the probability
of these outcomes must be 1.
00:04:24.680 --> 00:04:28.660
So we divide by the probability
of B. So we normalize.
00:04:28.660 --> 00:04:31.290
This now becomes-- the
probability of A given B
00:04:31.290 --> 00:04:36.310
is this share of B
weighted by the outcomes.
00:04:36.310 --> 00:04:36.810
OK.
00:04:39.020 --> 00:04:39.520
All right.
00:04:39.520 --> 00:04:45.410
For example then, what's the
probability of B given B?
00:04:47.970 --> 00:04:49.740
what's that equal?
00:04:49.740 --> 00:04:50.700
1.
00:04:50.700 --> 00:04:51.470
OK.
00:04:51.470 --> 00:04:54.406
Because we said it happened-- so
it happens with probability 1.
00:04:54.406 --> 00:04:58.890
Or, using the formula, that's
just probability of B and B
00:04:58.890 --> 00:05:01.890
divided by probability
of B. Well, that
00:05:01.890 --> 00:05:05.470
equals the probability
of B divided
00:05:05.470 --> 00:05:09.511
by the probability
of B, which is 1.
00:05:09.511 --> 00:05:10.010
All right.
00:05:10.010 --> 00:05:13.010
Any questions about
the definition
00:05:13.010 --> 00:05:15.970
of the conditional probability?
00:05:15.970 --> 00:05:16.500
Very simple.
00:05:16.500 --> 00:05:20.130
And it's easy to work
with using the formulas.
00:05:20.130 --> 00:05:22.850
Now, there's a nice
rule called the product
00:05:22.850 --> 00:05:25.635
rule, which follows from
the definition very simply.
00:05:31.600 --> 00:05:35.450
The product rule says that
the probability of A and B
00:05:35.450 --> 00:05:40.290
for two events is equal
to the probability of B
00:05:40.290 --> 00:05:44.880
times the probability
of A given B.
00:05:44.880 --> 00:05:47.060
And that's just follow
straightforwardly
00:05:47.060 --> 00:05:49.700
from this definition.
00:05:49.700 --> 00:05:53.750
Just multiply by probability
of B on both sides.
00:05:53.750 --> 00:05:54.250
All right.
00:05:54.250 --> 00:05:55.708
So now you have a
rule of computing
00:05:55.708 --> 00:05:59.750
a probability of two events
simultaneously happening.
00:05:59.750 --> 00:06:03.460
So for example, in the
Monty Hall problem,
00:06:03.460 --> 00:06:06.190
what's the probability
that Carol places
00:06:06.190 --> 00:06:11.840
the prize in box one and that's
the box the contestant chooses?
00:06:11.840 --> 00:06:12.340
All right?
00:06:12.340 --> 00:06:15.650
So if we took A and B
as defined up there,
00:06:15.650 --> 00:06:18.880
that's the probability that
Carol places it in box one
00:06:18.880 --> 00:06:22.090
and the contestant chose it.
00:06:22.090 --> 00:06:24.500
Well, that's the probability
that the contestant chooses
00:06:24.500 --> 00:06:28.740
it is 1/3 times the probability
that Carol put it there,
00:06:28.740 --> 00:06:33.130
given the contestant chose
it, or actually, vice versa,
00:06:33.130 --> 00:06:36.282
Is 1/9.
00:06:36.282 --> 00:06:38.430
OK?
00:06:38.430 --> 00:06:40.275
And this extends to more events.
00:06:42.870 --> 00:06:44.830
It is called the
general product rule.
00:06:55.500 --> 00:07:02.170
So if you want to compute
the probability of A1 and A2
00:07:02.170 --> 00:07:08.840
and all the way up to An, that's
simply the probability of a 1
00:07:08.840 --> 00:07:17.760
happening all by itself times
the probability of A2 given A1
00:07:17.760 --> 00:07:20.350
times-- well, I'll do the next
one-- times the probability
00:07:20.350 --> 00:07:29.750
of A3 given A1 and A2, dot,
dot dot, times, finally,
00:07:29.750 --> 00:07:32.940
the probability of An
given all the others.
00:07:40.730 --> 00:07:43.360
So that starts to look a
little more complicated.
00:07:43.360 --> 00:07:46.720
But it gives you a handy way
of computing the probability
00:07:46.720 --> 00:07:48.565
that an intersection
of events takes place.
00:07:51.080 --> 00:07:54.639
I do This is proved by induction
on n, just taking that rule
00:07:54.639 --> 00:07:55.680
and using induction on n.
00:07:55.680 --> 00:07:56.690
It's not hard.
00:07:56.690 --> 00:07:59.810
But we won't go through it.
00:07:59.810 --> 00:08:00.310
All right.
00:08:00.310 --> 00:08:02.770
Let's do some examples.
00:08:02.770 --> 00:08:04.070
We'll start with an easy one.
00:08:07.750 --> 00:08:11.670
Say you're playing
a playoff series
00:08:11.670 --> 00:08:13.430
and you're going to
play best 2 out of 3.
00:08:13.430 --> 00:08:15.358
All right.
00:08:15.358 --> 00:08:22.210
So you have a best
2 out of 3 series.
00:08:22.210 --> 00:08:25.350
So whoever wins the first two
games, best two out of three
00:08:25.350 --> 00:08:27.590
wins.
00:08:27.590 --> 00:08:32.210
And say you're told that
the probability of winning
00:08:32.210 --> 00:08:33.570
the first game is 1/2.
00:08:44.450 --> 00:08:48.560
So the teams are matched
50-50 for the first game.
00:08:48.560 --> 00:08:53.020
But then you're told that the
probability of winning a game
00:08:53.020 --> 00:08:55.590
after a victory is higher.
00:08:55.590 --> 00:08:56.165
It's 2/3.
00:09:02.000 --> 00:09:09.700
So the probability of
winning immediately
00:09:09.700 --> 00:09:22.190
after a game following
a win is two thirds.
00:09:22.190 --> 00:09:26.540
And similarly, the probability
of winning after a loss is 1/3.
00:09:43.370 --> 00:09:43.870
All right.
00:09:43.870 --> 00:09:46.000
And the idea here is
that you win a game,
00:09:46.000 --> 00:09:47.860
you're sort of psyched,
you've got momentum,
00:09:47.860 --> 00:09:50.870
and going into the next day
you're more likely to win.
00:09:50.870 --> 00:09:52.730
Similarly, if you lost
you're sort of down
00:09:52.730 --> 00:09:55.250
and the other guy has a
better chance of beating you.
00:09:55.250 --> 00:09:57.630
Now, what we're going
to try to figure out
00:09:57.630 --> 00:10:01.940
is the probability
of winning the series
00:10:01.940 --> 00:10:05.450
given you won the first game.
00:10:05.450 --> 00:10:05.950
All right?
00:10:05.950 --> 00:10:08.100
Now, conditional
probability comes up
00:10:08.100 --> 00:10:11.540
in two places in this problem.
00:10:11.540 --> 00:10:13.720
Anybody tell me places
where it's come up?
00:10:13.720 --> 00:10:15.970
So I got the problem statement
and the that's the goal
00:10:15.970 --> 00:10:17.870
is to figure out the probability
you win the series given
00:10:17.870 --> 00:10:18.828
you won the first game.
00:10:18.828 --> 00:10:21.030
So what's one place
conditional probability
00:10:21.030 --> 00:10:24.880
is entering into this problem?
00:10:24.880 --> 00:10:26.140
Yeah?
00:10:26.140 --> 00:10:27.550
AUDIENCE: The
probability changes
00:10:27.550 --> 00:10:31.252
depending on the result
of the previous game.
00:10:31.252 --> 00:10:32.210
PROFESSOR: That's true.
00:10:32.210 --> 00:10:34.270
The probability of winning
any particular game
00:10:34.270 --> 00:10:35.910
is influenced by
the previous game.
00:10:35.910 --> 00:10:38.780
So you're using conditional
probability there.
00:10:38.780 --> 00:10:40.140
All right.
00:10:40.140 --> 00:10:41.860
And where else?
00:10:41.860 --> 00:10:42.546
Yeah.
00:10:42.546 --> 00:10:46.434
AUDIENCE: [INAUDIBLE]
you have to take
00:10:46.434 --> 00:10:48.100
into account [INAUDIBLE].
00:10:48.100 --> 00:10:49.350
PROFESSOR: That's interesting.
00:10:49.350 --> 00:10:51.558
That will be another question
we're going to look at.
00:10:51.558 --> 00:10:54.050
What's the probability
of playing three games?
00:10:54.050 --> 00:10:55.340
Yep.
00:10:55.340 --> 00:10:57.521
That's one.
00:10:57.521 --> 00:10:58.020
OK.
00:10:58.020 --> 00:11:00.410
Well, the question
we're after, what's
00:11:00.410 --> 00:11:02.310
the probability of
winning the series given
00:11:02.310 --> 00:11:04.040
that you won the first game.
00:11:04.040 --> 00:11:07.040
We're going to compute a
conditional probability there.
00:11:07.040 --> 00:11:09.954
So it's coming up in a
couple of places here.
00:11:09.954 --> 00:11:10.870
All right.
00:11:10.870 --> 00:11:12.650
Let's figure this out.
00:11:12.650 --> 00:11:16.710
It's easy to do given
the tree method.
00:11:16.710 --> 00:11:19.610
So let's make the tree for this.
00:11:19.610 --> 00:11:27.300
So we have possibly three games
there's game one, game two,
00:11:27.300 --> 00:11:28.190
and game three.
00:11:31.060 --> 00:11:32.910
Game one, you can win or lose.
00:11:32.910 --> 00:11:36.560
There's two branches.
00:11:36.560 --> 00:11:38.135
Game two you can win or lose.
00:11:41.350 --> 00:11:46.650
And now, game three-- well, it
doesn't even take place here.
00:11:46.650 --> 00:11:47.450
But it does here.
00:11:47.450 --> 00:11:50.800
You can win or lose here.
00:11:50.800 --> 00:11:54.520
And you could win or lose here.
00:11:54.520 --> 00:11:56.120
And here the series is over.
00:11:56.120 --> 00:12:00.040
So there is no game
three in that case.
00:12:00.040 --> 00:12:03.030
The probabilities are
next we put a probability
00:12:03.030 --> 00:12:04.900
of every branch here.
00:12:04.900 --> 00:12:05.820
Game one is 50-50.
00:12:08.207 --> 00:12:10.040
What's the probability
you take this branch?
00:12:13.530 --> 00:12:17.170
2/3, because you're on the path
where you won the first game.
00:12:17.170 --> 00:12:18.760
You win the second
game with 2/3.
00:12:18.760 --> 00:12:21.540
You lose with 1/3.
00:12:21.540 --> 00:12:25.020
Now here you're on the path
where you lost the first game.
00:12:25.020 --> 00:12:29.931
So this has 1/3
and this has 2/3.
00:12:29.931 --> 00:12:30.430
All right?
00:12:30.430 --> 00:12:32.510
And then lastly, what's
the probability I have
00:12:32.510 --> 00:12:39.060
the win on the third game here?
00:12:39.060 --> 00:12:42.150
1/3, because I just
lost the last game.
00:12:42.150 --> 00:12:44.220
That's all I'm conditioning on.
00:12:44.220 --> 00:12:46.250
So that becomes 1/3.
00:12:46.250 --> 00:12:49.150
And this is 2/3 now.
00:12:49.150 --> 00:12:50.770
And then here I just won a game.
00:12:50.770 --> 00:12:54.386
So I've got 2/3 and 1/3.
00:12:54.386 --> 00:12:55.290
All right.
00:12:55.290 --> 00:12:57.890
So I got all the probabilities.
00:12:57.890 --> 00:13:02.870
And now I need to figure out
for the sample points what's
00:13:02.870 --> 00:13:03.730
their probability.
00:13:03.730 --> 00:13:07.760
So this sample point
we'll call win-win.
00:13:07.760 --> 00:13:11.176
This sample point
is win-lose-win.
00:13:15.089 --> 00:13:16.130
This one's win-lose-lose.
00:13:20.130 --> 00:13:27.780
Then we have lose-win-win,
lose-win-lose,
00:13:27.780 --> 00:13:30.040
and then lose-lose.
00:13:30.040 --> 00:13:32.010
So I got six sample points.
00:13:32.010 --> 00:13:35.430
And let's figure out the
probability for each one.
00:13:35.430 --> 00:13:39.280
Now remember the rule we
had for the tree method.
00:13:39.280 --> 00:13:42.570
I just multiply these things.
00:13:42.570 --> 00:13:44.820
Well, in fact, the
reason we have that rule
00:13:44.820 --> 00:13:49.120
is because that is the
same as the product rule.
00:13:49.120 --> 00:13:50.830
Because what I'm
asking here to compute
00:13:50.830 --> 00:14:03.210
the probability of this guy
is-- so the product rule gives
00:14:03.210 --> 00:14:07.700
the probability of a win-win
scenario-- win the first game,
00:14:07.700 --> 00:14:09.420
win the second game.
00:14:09.420 --> 00:14:11.210
By the product rule
is the probability
00:14:11.210 --> 00:14:16.580
that I win the first game
times the probability
00:14:16.580 --> 00:14:23.570
that I win the second game
given that I won the first game.
00:14:23.570 --> 00:14:27.280
That's what the
product rule says.
00:14:27.280 --> 00:14:31.590
Probability I win the
first game is 1/2 times
00:14:31.590 --> 00:14:33.480
the probability I
win the second given
00:14:33.480 --> 00:14:37.590
that I won the first is 2/3.
00:14:37.590 --> 00:14:40.270
So that equals 1/3.
00:14:45.641 --> 00:14:47.390
So what we're doing
here now is giving you
00:14:47.390 --> 00:14:51.270
the formal justification for
that rule that we had last time
00:14:51.270 --> 00:14:53.770
and that you'll always use--
is the probability of a sample
00:14:53.770 --> 00:14:56.320
point is the product
of the probabilities
00:14:56.320 --> 00:14:59.100
on the edges leading to it.
00:14:59.100 --> 00:15:00.310
It's just the product rule.
00:15:00.310 --> 00:15:03.260
Now the next
example is this one.
00:15:03.260 --> 00:15:06.155
And here we're going to use the
general product rule to get it.
00:15:09.190 --> 00:15:16.160
The probability of win-lose-win
by the general product rule
00:15:16.160 --> 00:15:20.440
is the probability that
you win the first game
00:15:20.440 --> 00:15:24.820
times the probability you
lose the second game given
00:15:24.820 --> 00:15:30.640
the that you win the first
times the probability you
00:15:30.640 --> 00:15:34.786
win the third given what?
00:15:34.786 --> 00:15:38.740
What am I given on
the product rule?
00:15:38.740 --> 00:15:43.582
Won the first, lost the second.
00:15:43.582 --> 00:15:44.419
All right.
00:15:44.419 --> 00:15:45.960
Well, now we can
fill in the numbers.
00:15:45.960 --> 00:15:49.768
The probability I win
the first is a 1/2.
00:15:49.768 --> 00:15:52.740
The probability that
I lose the second
00:15:52.740 --> 00:15:57.720
given that I won the
first, that's 1/3.
00:15:57.720 --> 00:15:59.440
And then this one
here, the probability
00:15:59.440 --> 00:16:01.680
that I win the third
given that I won the first
00:16:01.680 --> 00:16:04.600
and lost the second,
that simplifies
00:16:04.600 --> 00:16:09.160
the probability I win the third
given that I lost the second.
00:16:09.160 --> 00:16:11.795
Doesn't matter what
happened on the first.
00:16:11.795 --> 00:16:12.420
And that's 1/3.
00:16:16.010 --> 00:16:24.650
So this is 1/2 times
1/3 times 1/3 is 118.
00:16:24.650 --> 00:16:28.090
And that's 1/18.
00:16:28.090 --> 00:16:31.480
And it's just the product
because the product rule
00:16:31.480 --> 00:16:35.290
saying product of
the first probability
00:16:35.290 --> 00:16:37.910
times this one, which is
the conditional probability
00:16:37.910 --> 00:16:40.080
of being here times
this one, which
00:16:40.080 --> 00:16:45.600
is a conditional probability if
these events happened before.
00:16:45.600 --> 00:16:49.180
Any questions about that?
00:16:49.180 --> 00:16:51.240
Very simple to
do, which is good.
00:16:51.240 --> 00:16:51.740
Yeah.
00:16:51.740 --> 00:16:53.590
Is there a question?
00:16:53.590 --> 00:16:54.090
OK.
00:16:54.090 --> 00:16:54.714
All right.
00:16:54.714 --> 00:16:56.630
So let's fill in the
other probabilities here.
00:16:56.630 --> 00:16:59.500
I got 1/2, 1/3, and 2/3.
00:16:59.500 --> 00:17:01.820
That's 1/9.
00:17:01.820 --> 00:17:04.650
Same thing here is 1/9.
00:17:04.650 --> 00:17:08.670
This is 1/18 and 1/3.
00:17:12.069 --> 00:17:12.569
OK.
00:17:12.569 --> 00:17:15.319
So those are the probabilities
in the sample points.
00:17:15.319 --> 00:17:20.119
Now, to compute the probability
of winning the series given
00:17:20.119 --> 00:17:23.124
that we won the first game,
let's define the events here.
00:17:26.890 --> 00:17:32.355
So A be the event that
we win the series.
00:17:36.220 --> 00:17:42.420
B will be the event that
we win the first game.
00:17:45.580 --> 00:17:50.560
And I want to compute the
probability of A given B.
00:17:50.560 --> 00:17:53.020
And we use our formula.
00:17:53.020 --> 00:17:54.825
Where's the formula for that?
00:17:54.825 --> 00:17:56.170
It's way back over there.
00:17:56.170 --> 00:17:59.210
The probability of A
given B is the probability
00:17:59.210 --> 00:18:04.710
of both happening, the
probability of A and B
00:18:04.710 --> 00:18:09.450
divided by the probability of B.
00:18:09.450 --> 00:18:13.620
So now I just have to
compute these probabilities.
00:18:13.620 --> 00:18:16.700
So to do that I got to figure
out which sample points are
00:18:16.700 --> 00:18:20.820
in A and B here.
00:18:20.820 --> 00:18:22.030
So let's write that down.
00:18:22.030 --> 00:18:27.750
There's A, B, A
and B. All right.
00:18:27.750 --> 00:18:32.780
So A is the event that
we win the series.
00:18:35.670 --> 00:18:42.910
Now this sample point qualifies,
that one does, and this one.
00:18:42.910 --> 00:18:44.530
B is the event we
won the first game.
00:18:44.530 --> 00:18:48.360
And that's these
three sample points.
00:18:48.360 --> 00:18:54.521
And then A and B
intersect B is these two.
00:18:54.521 --> 00:18:55.020
All right.
00:18:55.020 --> 00:18:56.560
So for each event
that I care about
00:18:56.560 --> 00:18:58.684
I figure out which sample
points are in that event.
00:19:01.370 --> 00:19:05.570
And now I just add
the probabilities up.
00:19:05.570 --> 00:19:08.740
So what's the
probability of A and B?
00:19:17.400 --> 00:19:19.070
7/18.
00:19:19.070 --> 00:19:19.750
1/3 plus 1/18.
00:19:24.810 --> 00:19:26.165
What's the probability of B?
00:19:30.661 --> 00:19:31.160
Yeah.
00:19:31.160 --> 00:19:32.330
1/2, 9/18.
00:19:32.330 --> 00:19:33.430
I got these three points.
00:19:37.070 --> 00:19:43.770
So this'll be 1/3 third plus
1/18 plus the extra one, 1/9.
00:19:43.770 --> 00:19:49.340
So I've got 7/18 over 9/18.
00:19:49.340 --> 00:19:52.640
7/9 is the answer.
00:19:52.640 --> 00:19:54.470
So the probability
we win the Series
00:19:54.470 --> 00:19:56.370
given we won the
first game is 7/9.
00:19:59.032 --> 00:19:59.615
Any questions?
00:20:03.050 --> 00:20:06.090
We're going to do this same
thing about 10 different times.
00:20:06.090 --> 00:20:06.850
OK?
00:20:06.850 --> 00:20:09.170
And it will look a little
different each time maybe.
00:20:09.170 --> 00:20:10.250
But it's the same idea.
00:20:10.250 --> 00:20:13.780
And the beauty here is
it's really easy to do.
00:20:13.780 --> 00:20:16.400
I'm going to give you a
lot of confusing examples.
00:20:16.400 --> 00:20:20.696
But really, if you just do this
is it's going to be very easy.
00:20:20.696 --> 00:20:21.220
All right.
00:20:21.220 --> 00:20:24.670
Somebody talked about the
series lasting three games.
00:20:24.670 --> 00:20:27.722
What's the probability the
series lasts three games?
00:20:27.722 --> 00:20:29.712
Can anybody look at
that and tell me?
00:20:33.410 --> 00:20:37.310
1/3 because what you would do
is add up these three sample
00:20:37.310 --> 00:20:39.100
points.
00:20:39.100 --> 00:20:41.910
And it's the opposite
of these two.
00:20:41.910 --> 00:20:45.690
So it's 2/3 chance of two games,
a 1/3 chance of three games.
00:20:45.690 --> 00:20:49.910
So it's not likely
to go three games.
00:20:49.910 --> 00:20:51.640
All right.
00:20:51.640 --> 00:20:54.860
So to this point,
we've seen examples
00:20:54.860 --> 00:20:56.830
of a conditional
probability where
00:20:56.830 --> 00:21:03.570
it's A given B where A follows
B, like, we're told B happened.
00:21:03.570 --> 00:21:07.370
Now what's the chance of
A. And A is coming later.
00:21:07.370 --> 00:21:09.990
The probability of
winning today's game
00:21:09.990 --> 00:21:12.950
given that you won yesterday's
game, the probability
00:21:12.950 --> 00:21:17.770
of winning the series given
you already won the first game.
00:21:17.770 --> 00:21:21.140
Next, we're going to look
at the opposite scenario
00:21:21.140 --> 00:21:24.320
where the events are
reversed in order.
00:21:24.320 --> 00:21:28.240
The probability that
you won the first game
00:21:28.240 --> 00:21:32.294
given that you won the series.
00:21:32.294 --> 00:21:33.120
All right.
00:21:33.120 --> 00:21:37.140
Now, this is
inherently confusing
00:21:37.140 --> 00:21:40.170
because if you're
trying to figure--
00:21:40.170 --> 00:21:43.360
if you know you
the series, well,
00:21:43.360 --> 00:21:45.370
you already know what
happened in the first game
00:21:45.370 --> 00:21:47.270
because it's been played.
00:21:47.270 --> 00:21:49.270
So how could there be
any probability there?
00:21:49.270 --> 00:21:51.240
It happened.
00:21:51.240 --> 00:21:54.460
Well, so what the meaning
is is over all the times
00:21:54.460 --> 00:21:56.862
where the series
was played, sort
00:21:56.862 --> 00:21:58.570
of what fraction of
the time did the team
00:21:58.570 --> 00:22:00.760
that won the series win
the first game is one
00:22:00.760 --> 00:22:03.330
way you could think about it.
00:22:03.330 --> 00:22:05.120
Or, maybe you just don't know.
00:22:05.120 --> 00:22:06.024
The game was played.
00:22:06.024 --> 00:22:07.190
You know you won the series.
00:22:07.190 --> 00:22:09.300
But you don't know who
won the first game.
00:22:09.300 --> 00:22:12.540
And so you could think of a
probability still being there.
00:22:12.540 --> 00:22:17.220
Now when you think about it,
it gets me confused still.
00:22:17.220 --> 00:22:19.840
But just think about
it like the math.
00:22:19.840 --> 00:22:21.981
It's the same formula.
00:22:21.981 --> 00:22:22.480
OK.
00:22:22.480 --> 00:22:25.800
It doesn't matter which
happened first in time.
00:22:25.800 --> 00:22:27.750
You use the same mathematics.
00:22:27.750 --> 00:22:31.310
In fact, they give a special
name these kinds of things.
00:22:31.310 --> 00:22:34.480
They're called a postieri
conditional probabilities.
00:22:50.410 --> 00:22:55.060
It's a fancy name for just
saying that things are out
00:22:55.060 --> 00:22:57.711
of order in time.
00:22:57.711 --> 00:22:58.210
All right?
00:22:58.210 --> 00:23:07.920
So it's a probability of B given
A where B precedes A in time.
00:23:14.100 --> 00:23:16.150
All right?
00:23:16.150 --> 00:23:17.760
So it's the same math.
00:23:17.760 --> 00:23:21.500
It's just they're out of order.
00:23:21.500 --> 00:23:24.910
So let's figure
out the probability
00:23:24.910 --> 00:23:30.090
that you won the first game
given that you want the series.
00:23:30.090 --> 00:23:32.570
Let's figure it out.
00:23:32.570 --> 00:23:38.330
So I want probability of B
given A now for this example.
00:23:38.330 --> 00:23:42.620
Well, it's just the
probability of B and A
00:23:42.620 --> 00:23:47.590
over the probability of A.
00:23:47.590 --> 00:23:49.930
We already computed the
probability of A and B.
00:23:49.930 --> 00:23:50.950
That's 1/3 plus 1/18.
00:23:56.380 --> 00:23:58.760
what's the probability
of A, the probability
00:23:58.760 --> 00:24:00.290
of winning the first game?
00:24:04.705 --> 00:24:05.205
1/2.
00:24:05.205 --> 00:24:08.090
It's those three sample points
and they better add up to 1/2
00:24:08.090 --> 00:24:10.590
because we sort of said, the
probability of the first game's
00:24:10.590 --> 00:24:12.350
1/2.
00:24:12.350 --> 00:24:14.900
So that's over
1/2, which is 9/18.
00:24:17.780 --> 00:24:20.200
Well this was 7/18 over 9/18.
00:24:20.200 --> 00:24:22.980
It's 7/9.
00:24:22.980 --> 00:24:25.560
So the probability of
winning the first game given
00:24:25.560 --> 00:24:27.076
that you won series is 7/9.
00:24:29.950 --> 00:24:34.218
Anybody notice anything
unusual about that answer here?
00:24:34.218 --> 00:24:37.630
It's the same as the
answer over there.
00:24:37.630 --> 00:24:38.380
Is that a theorem?
00:24:40.970 --> 00:24:41.470
No.
00:24:41.470 --> 00:24:43.920
The probability of A
given B is not always
00:24:43.920 --> 00:24:47.230
the probability of B given
A. It was in this case.
00:24:47.230 --> 00:24:49.070
It is not always true.
00:24:49.070 --> 00:24:52.310
In fact, we could
make a simple example
00:24:52.310 --> 00:24:53.935
to see why that's
not always the case.
00:25:07.191 --> 00:25:07.690
All right.
00:25:07.690 --> 00:25:11.150
So say here's your sample space.
00:25:11.150 --> 00:25:15.850
And say that this
is B and this is
00:25:15.850 --> 00:25:20.990
A. What's the probability
of A given B in this case?
00:25:23.670 --> 00:25:24.330
1.
00:25:24.330 --> 00:25:25.980
If you're in B-- wait.
00:25:25.980 --> 00:25:26.480
No.
00:25:26.480 --> 00:25:28.850
It's not 1.
00:25:28.850 --> 00:25:32.820
What's the probability of
A given B If I got some--
00:25:32.820 --> 00:25:33.970
probably less than 1.
00:25:33.970 --> 00:25:37.210
Might be I've drawn it as
1/3 third if it was uniform.
00:25:37.210 --> 00:25:40.730
But in this case, the
probability of A given B
00:25:40.730 --> 00:25:41.580
is less than 1.
00:25:41.580 --> 00:25:43.450
What's the probability
of B given A?
00:25:47.240 --> 00:25:53.815
1, because if I'm in A I'm
definitely in B. All right.
00:25:53.815 --> 00:25:55.940
So that's an example where
they would be different.
00:25:55.940 --> 00:25:59.450
And that's the generic
case is they're different.
00:25:59.450 --> 00:26:01.840
All right?
00:26:01.840 --> 00:26:04.360
When are they equal because
they were equal in this case?
00:26:04.360 --> 00:26:06.640
What makes them equal?
00:26:06.640 --> 00:26:07.330
Let's see.
00:26:07.330 --> 00:26:11.900
When does the
probability of A given B
00:26:11.900 --> 00:26:14.170
equal a probability
of B given A?
00:26:14.170 --> 00:26:16.135
Let's see.
00:26:16.135 --> 00:26:19.150
Well, If I plug-in
the formula, this
00:26:19.150 --> 00:26:26.410
equals the probability of A and
B over the probability of B.
00:26:26.410 --> 00:26:35.000
That equals the probability of
B and A over a probability of A.
00:26:35.000 --> 00:26:36.030
So when are those equal?
00:26:39.430 --> 00:26:39.930
Yeah.
00:26:39.930 --> 00:26:43.380
When probability A equals
probability B. All right.
00:26:43.380 --> 00:26:45.070
So that's one case.
00:26:48.152 --> 00:26:49.310
What's the other case?
00:26:54.230 --> 00:26:55.520
Yeah-- when it's 0.
00:26:55.520 --> 00:26:57.460
Probability-- there's
no intersection.
00:26:57.460 --> 00:26:59.724
Probability of A
intersect B is 0.
00:26:59.724 --> 00:27:00.640
That's the other case.
00:27:07.071 --> 00:27:07.570
All right.
00:27:07.570 --> 00:27:10.030
But usually these
conditions won't
00:27:10.030 --> 00:27:13.690
apply-- just happened to in
this example by coincidence.
00:27:16.320 --> 00:27:21.750
Any questions about that?
00:27:21.750 --> 00:27:23.650
All right.
00:27:23.650 --> 00:27:24.150
Yeah.
00:27:24.150 --> 00:27:26.399
So the math is the same with
a postieri probabilities.
00:27:26.399 --> 00:27:29.210
It's really, really easy.
00:27:29.210 --> 00:27:29.830
All right.
00:27:29.830 --> 00:27:33.515
So let's do another simple
example that'll start to maybe
00:27:33.515 --> 00:27:34.640
be a little more confusing.
00:27:38.470 --> 00:27:40.590
Say we've got two coins.
00:27:53.510 --> 00:27:55.080
One of them is a fair coin.
00:27:59.200 --> 00:28:01.530
And by that, I mean the
probability comes up
00:28:01.530 --> 00:28:04.560
heads is the same as
the probability comes up
00:28:04.560 --> 00:28:06.980
tails is 1/2.
00:28:06.980 --> 00:28:08.770
The other one is an unfair coin.
00:28:12.440 --> 00:28:16.056
And in this case, that
means it's always heads.
00:28:16.056 --> 00:28:17.930
The probability of heads is 1.
00:28:17.930 --> 00:28:21.392
The probability of tails is 0.
00:28:21.392 --> 00:28:22.540
All right?
00:28:22.540 --> 00:28:25.370
I've got two such coins here.
00:28:25.370 --> 00:28:26.330
All right.
00:28:26.330 --> 00:28:30.870
Here is the unfair
coin-- heads and heads.
00:28:30.870 --> 00:28:33.480
Actually, they make these things
look like quarters sometimes.
00:28:33.480 --> 00:28:38.200
Here's the fair coin--
heads and tails.
00:28:38.200 --> 00:28:38.900
All right.
00:28:38.900 --> 00:28:45.410
Now suppose I pick one of
these at random, 50-50,
00:28:45.410 --> 00:28:50.590
I pick one of
these things, and I
00:28:50.590 --> 00:28:57.459
flip it, which I'm doing behind
my back, and lo and behold,
00:28:57.459 --> 00:28:58.875
it comes out and,
you see a heads.
00:29:02.240 --> 00:29:06.790
What's the probability
I'm holding the fair coin?
00:29:11.010 --> 00:29:15.700
I picked the coin,
50-50, behind my back.
00:29:15.700 --> 00:29:21.420
So one answer is, I picked the
fair coin with 50% probability.
00:29:21.420 --> 00:29:24.210
But then I flipped
it behind my back
00:29:24.210 --> 00:29:28.120
and I showed you the
result. And you see heads.
00:29:28.120 --> 00:29:31.250
Of course, if I'd
have shown you tails,
00:29:31.250 --> 00:29:33.850
You would have known for
sure it was the fair coin
00:29:33.850 --> 00:29:34.960
because that's the only
one with the tails.
00:29:34.960 --> 00:29:36.400
But you don't know for sure now.
00:29:36.400 --> 00:29:38.400
You see a heads.
00:29:38.400 --> 00:29:41.280
What's the probability this is
the fair coin given that you
00:29:41.280 --> 00:29:45.470
saw a heads after the flip?
00:29:45.470 --> 00:29:48.370
How many people think 1/2?
00:29:48.370 --> 00:29:50.940
After all, I picked it
with probability 1/2.
00:29:50.940 --> 00:29:54.890
How many people think
it's less than 1/2?
00:29:54.890 --> 00:29:55.680
Good.
00:29:55.680 --> 00:29:57.530
OK.
00:29:57.530 --> 00:30:00.110
Somebody even said 1/3.
00:30:00.110 --> 00:30:02.530
Does that sound right?
00:30:02.530 --> 00:30:04.870
A couple people like 1/3.
00:30:04.870 --> 00:30:07.160
OK.
00:30:07.160 --> 00:30:07.660
All right.
00:30:07.660 --> 00:30:09.990
Now, part of what
makes this tricky
00:30:09.990 --> 00:30:13.950
is I told you I picked the
coin with 50% probability.
00:30:13.950 --> 00:30:15.730
But then I gave you information.
00:30:15.730 --> 00:30:19.594
So I've conditioned the problem.
00:30:19.594 --> 00:30:21.010
And so this is one
of those things
00:30:21.010 --> 00:30:23.130
you could have an
ask Marilyn about.
00:30:23.130 --> 00:30:24.350
Is it 1/2 or is it 1/3?
00:30:24.350 --> 00:30:26.530
Because I picked
it with 50% chance,
00:30:26.530 --> 00:30:30.900
what does the
information do for you?
00:30:30.900 --> 00:30:34.190
Now, I'll give you a clue.
00:30:34.190 --> 00:30:37.660
Bobo might have written
in and said it's 1/2.
00:30:37.660 --> 00:30:39.784
And his proof is that
three other mathematicians
00:30:39.784 --> 00:30:40.450
agreed with him.
00:30:40.450 --> 00:30:41.640
[LAUGHTER]
00:30:41.640 --> 00:30:43.690
All right?
00:30:43.690 --> 00:30:44.550
OK.
00:30:44.550 --> 00:30:48.030
So let's figure it out.
00:30:48.030 --> 00:30:50.320
And really it's very simple.
00:30:50.320 --> 00:30:54.550
It's just drawing out
the tree and computing
00:30:54.550 --> 00:30:56.480
the conditional probability.
00:30:56.480 --> 00:31:00.070
So we're going to do the same
thing over and over again
00:31:00.070 --> 00:31:03.464
because it just works
for every problem.
00:31:03.464 --> 00:31:07.950
Of course, you could imagine
debating this for awhile,
00:31:07.950 --> 00:31:08.890
arguing with somebody.
00:31:08.890 --> 00:31:10.200
Is it 1/2 or 1/3?
00:31:10.200 --> 00:31:14.439
Much simpler just to do it.
00:31:14.439 --> 00:31:16.605
So the first thing is we
have, which coin is picked?
00:31:19.300 --> 00:31:22.040
So it could be
fair-- and I told you
00:31:22.040 --> 00:31:26.650
that happens with
probability 1/2-- or unfair,
00:31:26.650 --> 00:31:29.300
which is also 1/2.
00:31:29.300 --> 00:31:33.290
Then we have the flip.
00:31:33.290 --> 00:31:35.570
The fair coin is
equally likely to be
00:31:35.570 --> 00:31:39.170
heads or tails, each with 1/2.
00:31:39.170 --> 00:31:46.190
The unfair coin, guaranteed
to be heads, probability 1.
00:31:46.190 --> 00:31:46.730
All right.
00:31:46.730 --> 00:31:50.510
Now we get the sample
point outcomes.
00:31:50.510 --> 00:31:54.310
It's fair in heads with
the probability 1/4,
00:31:54.310 --> 00:31:57.990
fair in tails,
probability 1/4, unfair
00:31:57.990 --> 00:32:01.580
in heads, probability 1/2.
00:32:01.580 --> 00:32:05.070
Now we define the
events of interest.
00:32:05.070 --> 00:32:07.510
A is going to be that
we chose the fair coin.
00:32:13.670 --> 00:32:16.450
And B is at the
result, is heads.
00:32:19.980 --> 00:32:22.150
And of course what
I want to know
00:32:22.150 --> 00:32:25.640
is the probability that I
chose the fair coin given
00:32:25.640 --> 00:32:26.660
that I saw a heads.
00:32:29.610 --> 00:32:32.020
So to do that we
plug in our formula.
00:32:32.020 --> 00:32:37.750
That's just the
probability of A and B
00:32:37.750 --> 00:32:42.190
over the probability
of B. And to compute
00:32:42.190 --> 00:32:45.890
that I got to figure out
the probability of A and B
00:32:45.890 --> 00:32:47.240
and the probability of B.
00:32:47.240 --> 00:32:49.110
So I'll make my diagram.
00:32:49.110 --> 00:32:54.970
A here, B here, A
and B. A is the event
00:32:54.970 --> 00:32:57.570
I chose the fair coin.
00:32:57.570 --> 00:33:00.050
That's these guys.
00:33:00.050 --> 00:33:02.850
B is the event the
result is heads.
00:33:02.850 --> 00:33:05.490
That's this one and this one.
00:33:08.070 --> 00:33:12.730
And A intersect B,
That's the only point.
00:33:12.730 --> 00:33:16.450
So this is really
easy to compute now.
00:33:16.450 --> 00:33:18.050
What's the probability
of A and B?
00:33:20.770 --> 00:33:21.490
1/4.
00:33:21.490 --> 00:33:24.270
It's just that sample point.
00:33:24.270 --> 00:33:25.865
What's the probability of B?
00:33:28.400 --> 00:33:30.480
3/4, 1/4 plus 1/2.
00:33:33.100 --> 00:33:35.630
So the probability
of A given B is 1/3.
00:33:38.650 --> 00:33:41.400
Really simple to
answer this question.
00:33:41.400 --> 00:33:43.300
Just don't even think about it.
00:33:43.300 --> 00:33:46.460
Just write down the tree
when you get these things.
00:33:46.460 --> 00:33:50.752
So much easier just to
write the tree down.
00:33:50.752 --> 00:33:51.720
All right.
00:33:51.720 --> 00:33:57.690
Now the key here is we
knew the probability
00:33:57.690 --> 00:34:01.445
of picking the fair
coin in the first place.
00:34:01.445 --> 00:34:03.090
Maybe it's worth
writing down what
00:34:03.090 --> 00:34:07.268
happens if that's a variable--
sum variable P. Let's do that.
00:34:15.239 --> 00:34:20.130
For example, what if I hadn't
told you the probability
00:34:20.130 --> 00:34:22.690
that I picked the fair coin?
00:34:22.690 --> 00:34:26.420
I just picked one
and flipped it.
00:34:26.420 --> 00:34:30.130
Think that'll change the answer?
00:34:30.130 --> 00:34:32.940
It should because you got
to plug something in there
00:34:32.940 --> 00:34:34.927
for the 1/2 for this to work.
00:34:34.927 --> 00:34:36.010
So let's see what happens.
00:34:36.010 --> 00:34:40.889
Say I picked the fair
coin with probability
00:34:40.889 --> 00:34:46.880
P and the unfair
coin with 1 minus P.
00:34:46.880 --> 00:34:52.429
And this is the same
heads and tails, 1/2, 1/2.
00:34:52.429 --> 00:34:56.239
Heads, the probability 1.
00:34:56.239 --> 00:35:01.520
Well now, instead of 1/4
I get P over 2 up here.
00:35:01.520 --> 00:35:03.960
And this is now 1
minus P instead of 1/2.
00:35:06.580 --> 00:35:12.736
So the probability of A given
B is the probability of A and B
00:35:12.736 --> 00:35:16.220
is p over 2.
00:35:16.220 --> 00:35:19.890
And the probability
of B is P over 2
00:35:19.890 --> 00:35:31.100
plus 1 minus P. That's P over
2 up top, one minus P over 2,
00:35:31.100 --> 00:35:33.950
and that is all
multiplied by-- what am I
00:35:33.950 --> 00:35:35.720
going to multiply-- 2 here.
00:35:35.720 --> 00:35:41.210
I'll get P over 2 minus P.
00:35:41.210 --> 00:35:44.810
So the probability with which
I picked the coin to start with
00:35:44.810 --> 00:35:48.060
impacts the answer here.
00:35:48.060 --> 00:35:55.680
For example, what if I picked
the unfair coin for sure?
00:35:55.680 --> 00:35:57.330
That would be P being 0.
00:36:00.820 --> 00:36:03.130
Well, the probability that
I picked the fair coin
00:36:03.130 --> 00:36:06.704
is 0 over 2, which is 0.
00:36:06.704 --> 00:36:08.870
All right though-- even
know I showed you the heads,
00:36:08.870 --> 00:36:12.260
there's no chance it was the
fair coin because I picked
00:36:12.260 --> 00:36:13.610
the unfair coin for sure.
00:36:19.840 --> 00:36:23.020
Same thing if I picked
the fair coin for sure,
00:36:23.020 --> 00:36:25.110
better be the case this is 1.
00:36:25.110 --> 00:36:26.720
So I get 1 over 2 minus 1.
00:36:26.720 --> 00:36:27.400
It's 1.
00:36:30.380 --> 00:36:32.610
Any questions?
00:36:32.610 --> 00:36:36.044
So it's important you know
the probability I picked
00:36:36.044 --> 00:36:37.210
the fair coin to start with.
00:36:37.210 --> 00:36:39.630
Otherwise, you
can't go anywhere.
00:36:42.700 --> 00:36:43.930
All right.
00:36:43.930 --> 00:36:46.640
What if I do the same game?
00:36:46.640 --> 00:36:50.910
Pick a coin with probability p.
00:36:50.910 --> 00:36:54.291
But now I flip it K times.
00:36:54.291 --> 00:36:56.400
Say I flip it 100 times.
00:36:56.400 --> 00:36:58.340
And every time it
comes up heads.
00:37:01.420 --> 00:37:04.720
I mean you're pretty sure you
got the unfair coin because you
00:37:04.720 --> 00:37:05.850
never saw a tails.
00:37:05.850 --> 00:37:07.480
Right?
00:37:07.480 --> 00:37:08.220
So let's do that.
00:37:08.220 --> 00:37:11.060
Let's compute that scenario.
00:37:11.060 --> 00:37:16.430
So instead of a single
heads I get K straight heads
00:37:16.430 --> 00:37:19.310
and no tails.
00:37:19.310 --> 00:37:22.290
This would happen with
1 over 2 to the K.
00:37:22.290 --> 00:37:27.270
This would happen with 1
minus 1 over 2 to the K.
00:37:27.270 --> 00:37:31.980
So this is now p over 2 to
the K. This is now P1 minus 2
00:37:31.980 --> 00:37:34.870
to the minus K.
00:37:34.870 --> 00:37:36.380
Let's recompute
the probabilities.
00:37:36.380 --> 00:37:38.657
I'm going somewhere where this.
00:37:38.657 --> 00:37:39.240
Wait a minute.
00:37:50.340 --> 00:37:51.800
So now we're
looking at the event
00:37:51.800 --> 00:37:56.255
that B is K straight heads.
00:37:59.080 --> 00:38:02.150
Come up.
00:38:02.150 --> 00:38:04.240
And I want to know
the probability
00:38:04.240 --> 00:38:07.420
that I picked the fair coin
given that it just never comes
00:38:07.420 --> 00:38:09.700
up tails.
00:38:09.700 --> 00:38:10.575
The math is the same.
00:38:17.390 --> 00:38:19.895
The probability now that
I picked the fair coin
00:38:19.895 --> 00:38:23.310
and got k straight
heads is just p times 2
00:38:23.310 --> 00:38:31.490
to the minus K. The probability
that I got K straight heads is
00:38:31.490 --> 00:38:34.210
P times 2 to the minus
K plus the chance
00:38:34.210 --> 00:38:38.230
I picked the unfair
coin, which is 1 minus P.
00:38:38.230 --> 00:38:40.340
And if I multiply top
and bottom by 2 to the K,
00:38:40.340 --> 00:38:47.571
I get P over P plus
to the K 1 minus B.
00:38:47.571 --> 00:38:48.070
All right.
00:38:48.070 --> 00:38:53.290
So it gets very unlikely that
I've got the fair coin here
00:38:53.290 --> 00:38:55.280
as K gets big.
00:38:55.280 --> 00:38:58.660
Like if K is 100 I got
a big number down here.
00:38:58.660 --> 00:39:00.680
And basically it's
0 chance-- close
00:39:00.680 --> 00:39:03.900
to 0 chance of the fair coin.
00:39:06.910 --> 00:39:10.760
But now say I do the
following experiment.
00:39:10.760 --> 00:39:16.820
I don't tell you P.
But I pull a coin out
00:39:16.820 --> 00:39:21.380
and 100 flips in
a row it's heads.
00:39:21.380 --> 00:39:22.710
Which coin do you think I have?
00:39:26.340 --> 00:39:29.300
I flipped it 100 straight times
and it's heads every time.
00:39:32.550 --> 00:39:33.050
Yeah.
00:39:33.050 --> 00:39:34.341
There's not enough information.
00:39:34.341 --> 00:39:34.990
You don't know.
00:39:34.990 --> 00:39:37.840
What do you want to say?
00:39:37.840 --> 00:39:40.890
You want to say
it's the unfair coin
00:39:40.890 --> 00:39:46.250
but you have no idea because I
might have picked the fair coin
00:39:46.250 --> 00:39:48.790
with probability 1, in which
case it is the fair coin
00:39:48.790 --> 00:39:51.000
and it just was unlucky
that it came up heads
00:39:51.000 --> 00:39:53.270
100 times in a row.
00:39:53.270 --> 00:39:54.530
But it could be.
00:39:54.530 --> 00:39:58.730
So you could say nothing if you
don't know the probability P.
00:39:58.730 --> 00:40:05.370
Because sure enough, if
I plug in P being 1 here,
00:40:05.370 --> 00:40:07.850
that wipes out the 2 to the K
and I just get probability 1.
00:40:10.600 --> 00:40:12.940
OK?
00:40:12.940 --> 00:40:13.440
All right.
00:40:13.440 --> 00:40:18.510
Now when this comes
up in practice
00:40:18.510 --> 00:40:20.560
is with things like polling.
00:40:20.560 --> 00:40:22.680
Like, we just had an election.
00:40:22.680 --> 00:40:25.430
And people do poles
ahead of time.
00:40:25.430 --> 00:40:28.850
And they sample
thousands of voters
00:40:28.850 --> 00:40:31.160
from 1% of the population.
00:40:31.160 --> 00:40:33.590
And they say, OK,
that 60% of the people
00:40:33.590 --> 00:40:37.240
are going to vote Republican.
00:40:37.240 --> 00:40:39.880
And they might have a margin of
error, three points, whatever
00:40:39.880 --> 00:40:40.380
that means.
00:40:40.380 --> 00:40:42.580
And we'll figure
that out next week.
00:40:42.580 --> 00:40:45.920
What does that tell you about
the electorate as a whole--
00:40:45.920 --> 00:40:52.180
the population if they sample 1%
at random, 60% are Republican.
00:40:52.180 --> 00:40:54.572
Yeah?
00:40:54.572 --> 00:40:58.476
AUDIENCE: [INAUDIBLE]
The options you have,
00:40:58.476 --> 00:41:00.428
is it all heads or
is it all tails?
00:41:00.428 --> 00:41:01.892
It should be one
option all heads
00:41:01.892 --> 00:41:03.860
and another option
at least one tails.
00:41:03.860 --> 00:41:04.910
PROFESSOR: You're right.
00:41:04.910 --> 00:41:06.320
Oops.
00:41:06.320 --> 00:41:06.940
All right.
00:41:06.940 --> 00:41:07.903
At least one tail for this one.
00:41:07.903 --> 00:41:08.402
Yeah.
00:41:08.402 --> 00:41:10.168
Good.
00:41:10.168 --> 00:41:11.530
That is true.
00:41:11.530 --> 00:41:13.870
OK.
00:41:13.870 --> 00:41:18.550
Any questions
about that example?
00:41:18.550 --> 00:41:19.550
OK.
00:41:19.550 --> 00:41:21.300
Now we're back to the
election and there's
00:41:21.300 --> 00:41:26.520
a pole that says they sampled
1% of the population at random
00:41:26.520 --> 00:41:29.540
and 60% said they're
going to vote Republican.
00:41:29.540 --> 00:41:32.570
And the margin of error
is 3% or something.
00:41:32.570 --> 00:41:36.510
What does that tell you about
the population of the country?
00:41:36.510 --> 00:41:37.900
Nothing.
00:41:37.900 --> 00:41:39.560
That's right.
00:41:39.560 --> 00:41:42.490
It is what it is.
00:41:42.490 --> 00:41:47.200
All you can conclude is
that either the population
00:41:47.200 --> 00:41:51.920
is close to 60% Republican
or you were unlucky
00:41:51.920 --> 00:41:54.740
in the 1% you sample.
00:41:54.740 --> 00:41:59.120
That's what you can conclude
because the population really
00:41:59.120 --> 00:42:00.110
is fixed in this case.
00:42:00.110 --> 00:42:01.060
It is what it is.
00:42:01.060 --> 00:42:04.090
There's no randomness
in the population.
00:42:04.090 --> 00:42:04.590
All right?
00:42:04.590 --> 00:42:07.630
So you have next
week for recitation.
00:42:07.630 --> 00:42:09.890
You're going to design a
pole and work through how
00:42:09.890 --> 00:42:11.640
to calculate the margin
of error and work
00:42:11.640 --> 00:42:14.640
through what that
really means in terms
00:42:14.640 --> 00:42:17.410
of what the population is like.
00:42:17.410 --> 00:42:20.530
Now of course, if it comes
out 100 straight times heads,
00:42:20.530 --> 00:42:23.590
you've got to be really
unlucky to have the fair coin.
00:42:23.590 --> 00:42:25.460
And the same thing
with designing the poll
00:42:25.460 --> 00:42:28.270
if you're way off.
00:42:28.270 --> 00:42:32.761
Any questions about that?
00:42:32.761 --> 00:42:33.260
OK.
00:42:33.260 --> 00:42:37.970
The next example comes up
all the time in practice.
00:42:37.970 --> 00:42:40.946
And that's with medical testing.
00:42:40.946 --> 00:42:43.547
Maybe I'll leave-- no.
00:42:43.547 --> 00:42:44.380
I'll take that down.
00:42:44.380 --> 00:42:45.634
We know that now.
00:43:05.030 --> 00:43:08.520
Now in this case--
in fact, this is
00:43:08.520 --> 00:43:11.530
a question we had on the
final exam a few years ago.
00:43:11.530 --> 00:43:14.230
And there's a good chance
this kind of question's
00:43:14.230 --> 00:43:16.340
going to be on the
final this year.
00:43:16.340 --> 00:43:18.760
There's a disease out there.
00:43:18.760 --> 00:43:21.390
And you can have a test for it.
00:43:21.390 --> 00:43:24.507
But like most medical
tests, they're not perfect.
00:43:24.507 --> 00:43:26.840
Sometimes when it says you've
got the disease you really
00:43:26.840 --> 00:43:27.900
don't.
00:43:27.900 --> 00:43:31.657
And if it ways you don't
have it, you really do.
00:43:31.657 --> 00:43:33.240
So in this case,
we're going to assume
00:43:33.240 --> 00:43:45.914
that 10% of the population has
the disease, whatever it is.
00:43:45.914 --> 00:43:47.330
You don't get
symptoms right away.
00:43:47.330 --> 00:43:49.190
So you have this test.
00:43:49.190 --> 00:44:04.142
But if you have the disease
there is a 10% chance
00:44:04.142 --> 00:44:05.225
that the test is negative.
00:44:08.940 --> 00:44:15.040
And this is called
a false negative,
00:44:15.040 --> 00:44:17.410
because the test comes back
negative but it's wrong,
00:44:17.410 --> 00:44:19.750
because you have the disease.
00:44:19.750 --> 00:44:22.620
And similarly, if you
have the disease--
00:44:22.620 --> 00:44:29.240
or sorry-- if you
don't have the disease,
00:44:29.240 --> 00:44:35.170
there's a 30% chance that
the test comes back positive.
00:44:35.170 --> 00:44:41.560
And it's called a false positive
because it came back positive,
00:44:41.560 --> 00:44:43.870
but you don't have it.
00:44:43.870 --> 00:44:45.800
So the test is pretty good.
00:44:45.800 --> 00:44:46.500
Right?
00:44:46.500 --> 00:44:52.390
It's 10% false negative right,
30% false positive right.
00:44:52.390 --> 00:44:59.210
Now say you select a random
person and they test positive.
00:44:59.210 --> 00:45:01.120
What you want to know
is the probability
00:45:01.120 --> 00:45:04.014
they have the disease given
that it's a random person.
00:45:07.730 --> 00:45:12.630
So actually, this came
up in my personal life.
00:45:12.630 --> 00:45:18.080
Many years ago when my wife
was pregnant with Alex,
00:45:18.080 --> 00:45:22.840
she was exposed to somebody
with TB here at MIT.
00:45:22.840 --> 00:45:25.400
And she took the test.
00:45:25.400 --> 00:45:28.040
And it came back positive.
00:45:28.040 --> 00:45:30.639
Now the bad thing--
TB's a bad thing.
00:45:30.639 --> 00:45:31.680
You don't want to get it.
00:45:31.680 --> 00:45:35.990
But the medicine for it
you take for six months.
00:45:35.990 --> 00:45:37.760
And she was worried
about taking medicine
00:45:37.760 --> 00:45:40.530
for six months when she's
pregnant because who
00:45:40.530 --> 00:45:43.830
knows what the TB medicine
does kind of thing
00:45:43.830 --> 00:45:45.940
if you have a baby.
00:45:45.940 --> 00:45:48.440
So she asked the doc, what's
the probability I really
00:45:48.440 --> 00:45:51.320
have the disease?
00:45:51.320 --> 00:45:53.570
The doc doesn't know.
00:45:53.570 --> 00:45:55.620
The doc maybe could give
you some of these steps,
00:45:55.620 --> 00:45:58.350
10% false negative,
30% false positive.
00:45:58.350 --> 00:45:59.760
But it tested positive.
00:45:59.760 --> 00:46:03.530
So they just normally
give you the medicine.
00:46:03.530 --> 00:46:05.140
So say this was the story.
00:46:05.140 --> 00:46:06.490
What would you say?
00:46:06.490 --> 00:46:07.670
What do you think?
00:46:07.670 --> 00:46:10.090
How many people think that
it's a least a 70% chance
00:46:10.090 --> 00:46:12.900
you got the disease?
00:46:12.900 --> 00:46:14.930
She tested positive
and it's only
00:46:14.930 --> 00:46:16.820
got a 30% false positive rate.
00:46:16.820 --> 00:46:18.701
Anybody?
00:46:18.701 --> 00:46:21.260
So you don't think
she's likely to have it.
00:46:21.260 --> 00:46:23.720
How many people think
it's better than 50-50
00:46:23.720 --> 00:46:26.820
you have the disease?
00:46:26.820 --> 00:46:27.870
A few.
00:46:27.870 --> 00:46:31.420
How many people
think less than 50%.
00:46:31.420 --> 00:46:32.290
A bunch.
00:46:32.290 --> 00:46:33.070
Yeah.
00:46:33.070 --> 00:46:36.250
You're right, in fact.
00:46:36.250 --> 00:46:37.440
Let's figure out the answer.
00:46:37.440 --> 00:46:39.550
It's easy to do.
00:46:39.550 --> 00:46:43.935
So A is the event the
person has the disease.
00:46:51.800 --> 00:46:56.835
And B is the event that
the person tests positive.
00:47:01.120 --> 00:47:03.400
And of course what
we want to know
00:47:03.400 --> 00:47:05.670
is the probability you
have the disease given
00:47:05.670 --> 00:47:07.550
that you tested positive.
00:47:07.550 --> 00:47:11.420
And that's just the probability
of both events divided
00:47:11.420 --> 00:47:16.340
by the probability
of testing positive.
00:47:16.340 --> 00:47:20.060
So let's figure that
out by drawing the tree.
00:47:38.750 --> 00:47:40.970
So first, do you
have the disease?
00:47:43.980 --> 00:47:45.360
And it's yes or no.
00:47:49.320 --> 00:47:50.380
And let's see.
00:47:50.380 --> 00:47:53.250
The probability of
having the disease, what
00:47:53.250 --> 00:47:56.010
is that for a random person?
00:47:56.010 --> 00:47:56.810
10%.
00:47:56.810 --> 00:47:58.380
that the stat.
00:47:58.380 --> 00:48:01.274
So it's-- actually,
we'll call it 0.1.
00:48:01.274 --> 00:48:03.997
And 9.9 you don't have it.
00:48:03.997 --> 00:48:05.080
And then there's the test.
00:48:07.940 --> 00:48:09.995
Well, you can be
positive or negative.
00:48:14.360 --> 00:48:19.520
Now if you have
the disease, there
00:48:19.520 --> 00:48:26.880
is a-- the chance you
test negative is 10%, 0.1.
00:48:26.880 --> 00:48:31.352
Therefore there's a 90%
chance you test positive.
00:48:31.352 --> 00:48:32.810
Now if, you don't
have the disease,
00:48:32.810 --> 00:48:33.893
you could test either way.
00:48:36.740 --> 00:48:40.390
If you don't have the
disease there's a 30% chance
00:48:40.390 --> 00:48:42.590
you test positive.
00:48:42.590 --> 00:48:47.760
30 here and 70% percent
chance you're negative.
00:48:47.760 --> 00:48:51.620
Now we can compute each
sample point probability.
00:48:51.620 --> 00:48:56.560
This one is 0.1
times 0.9 is 0.09.
00:48:56.560 --> 00:49:00.050
0.1 times 1 is 0.01.
00:49:00.050 --> 00:49:03.720
0.9 and 0.3 is 0.27.
00:49:03.720 --> 00:49:09.580
0.9 and 0.7 is 0.63.
00:49:09.580 --> 00:49:12.350
So all sample points
are figured out.
00:49:12.350 --> 00:49:16.240
Now we figure out which sample
points are in which sets.
00:49:16.240 --> 00:49:21.650
So we have event A, event B,
and A intersect B. Let's see.
00:49:21.650 --> 00:49:25.300
A is the event you
have the disease.
00:49:25.300 --> 00:49:27.590
That's these guys.
00:49:27.590 --> 00:49:31.790
B is the event
you test positive.
00:49:31.790 --> 00:49:36.840
That's this one and this one.
00:49:36.840 --> 00:49:40.521
A intersect B is just this one.
00:49:40.521 --> 00:49:41.020
All right.
00:49:41.020 --> 00:49:43.149
We're almost done.
00:49:43.149 --> 00:49:44.690
Let's just figure
out the probability
00:49:44.690 --> 00:49:47.190
you have the disease.
00:49:47.190 --> 00:49:49.000
What's the probability
of A intersect B?
00:49:52.340 --> 00:49:53.120
0.09.
00:49:53.120 --> 00:49:56.740
It's just that one sample point.
00:49:56.740 --> 00:49:59.645
What's the probability
that you tested positive?
00:50:02.170 --> 00:50:03.230
0.36.
00:50:03.230 --> 00:50:03.730
Yeah.
00:50:03.730 --> 00:50:13.610
0.09 plus 0.27, which is 0.36.
00:50:13.610 --> 00:50:21.430
So I got 0.09 over 0.36 is 1/4.
00:50:21.430 --> 00:50:22.790
Wow.
00:50:22.790 --> 00:50:24.071
That seems bizarre.
00:50:24.071 --> 00:50:24.570
Right?
00:50:24.570 --> 00:50:28.740
You've got a test, 10%
percent false negative,
00:50:28.740 --> 00:50:30.250
30% false positive.
00:50:30.250 --> 00:50:34.060
Yet, when you test positive
there's only a 25% chance
00:50:34.060 --> 00:50:36.490
you have the disease.
00:50:36.490 --> 00:50:38.610
So maybe you don't
take the medicine.
00:50:38.610 --> 00:50:40.830
So if there's risk
both ways, probably
00:50:40.830 --> 00:50:42.930
don't have the disease.
00:50:42.930 --> 00:50:44.970
Yeah?
00:50:44.970 --> 00:50:46.470
AUDIENCE: [INAUDIBLE]
disease change
00:50:46.470 --> 00:50:47.970
because you've
already been exposed
00:50:47.970 --> 00:50:48.970
to somebody that has it?
00:50:48.970 --> 00:50:52.530
PROFESSOR: That's a
great point, great point,
00:50:52.530 --> 00:50:54.440
because there's
additional information
00:50:54.440 --> 00:50:57.190
conditioning this in the
personal example I cited.
00:50:57.190 --> 00:50:59.250
You were exposed to somebody.
00:50:59.250 --> 00:51:02.030
So we need to condition
on that as well, which
00:51:02.030 --> 00:51:04.250
raises the chance
you have the disease.
00:51:04.250 --> 00:51:06.040
That's a great point.
00:51:06.040 --> 00:51:07.950
Yeah.
00:51:07.950 --> 00:51:10.420
Just like in the-- well, we
haven't got to that example.
00:51:10.420 --> 00:51:12.420
Do another example with
that exact kind of thing
00:51:12.420 --> 00:51:14.431
is very important.
00:51:14.431 --> 00:51:14.930
All right.
00:51:14.930 --> 00:51:17.920
So this is sort of
paradoxical that it
00:51:17.920 --> 00:51:21.120
looks like a pretty good
test-- low false positive,
00:51:21.120 --> 00:51:25.460
full false negatives, but
likely be wrong, at least if it
00:51:25.460 --> 00:51:28.637
tells you have the disease.
00:51:28.637 --> 00:51:29.720
In fact, let's figure out.
00:51:29.720 --> 00:51:31.720
What's the probability
that the test is correct?
00:51:35.620 --> 00:51:39.130
What's the probability the
test is right in general?
00:51:43.030 --> 00:51:44.350
72%.
00:51:44.350 --> 00:51:45.570
Let's see.
00:51:45.570 --> 00:51:48.731
So it would be 0.09 plus 0.63.
00:51:48.731 --> 00:51:49.231
72%.
00:51:56.780 --> 00:51:59.704
So it's likely to be right.
00:51:59.704 --> 00:52:01.370
But if it tells you
you have the disease
00:52:01.370 --> 00:52:02.460
it's likely to be wrong.
00:52:05.500 --> 00:52:06.850
It's hard.
00:52:06.850 --> 00:52:08.530
Why is this happening?
00:52:08.530 --> 00:52:09.780
Why does it come out that way?
00:52:09.780 --> 00:52:10.280
Yeah?
00:52:12.980 --> 00:52:16.190
AUDIENCE: Then there is
only a 1 in 64 chance
00:52:16.190 --> 00:52:18.630
that you have the disease.
00:52:18.630 --> 00:52:20.790
So if it comes back
negative, then it's
00:52:20.790 --> 00:52:22.584
a pretty good indication
that you're OK.
00:52:22.584 --> 00:52:23.250
PROFESSOR: Yeah.
00:52:23.250 --> 00:52:26.980
If it comes back negative than
it really is doing very well.
00:52:26.980 --> 00:52:27.907
That's right.
00:52:27.907 --> 00:52:29.615
But why is it when it
comes back positive
00:52:29.615 --> 00:52:32.156
that you're unlikely to have
the disease if it's a good test.
00:52:32.156 --> 00:52:32.886
Yeah.
00:52:32.886 --> 00:52:34.260
AUDIENCE: The
disease is so rare.
00:52:34.260 --> 00:52:35.760
PROFESSOR: The
disease is so rare.
00:52:35.760 --> 00:52:37.500
Absolutely.
00:52:37.500 --> 00:52:39.770
This number here is so small.
00:52:39.770 --> 00:52:41.960
And that's what's doing it.
00:52:41.960 --> 00:52:45.020
Because if you look at how
many people have the disease
00:52:45.020 --> 00:52:47.120
and test positive, it's 0.09.
00:52:47.120 --> 00:52:50.270
So many people don't
have the disease
00:52:50.270 --> 00:52:53.120
that even with a small false
positive rate, this number
00:52:53.120 --> 00:52:55.380
swamps out that number.
00:52:55.380 --> 00:52:57.930
In fact, imagine
nobody had the disease.
00:52:57.930 --> 00:53:00.420
You'd have a 0 here.
00:53:00.420 --> 00:53:01.140
All right?
00:53:01.140 --> 00:53:04.750
And then you would always be
wrong if you said you had it.
00:53:04.750 --> 00:53:05.480
OK?
00:53:05.480 --> 00:53:08.780
That's good.
00:53:08.780 --> 00:53:09.280
OK.
00:53:09.280 --> 00:53:11.980
This comes up in weather
prediction, the same paradox.
00:53:11.980 --> 00:53:14.050
For example, say you're
trying to predict
00:53:14.050 --> 00:53:15.750
the weather for Seattle.
00:53:15.750 --> 00:53:18.160
Sometimes it seems
like this in Boston.
00:53:18.160 --> 00:53:22.490
And you just say,
it's going to rain.
00:53:22.490 --> 00:53:25.272
Forget all the fancy weather
forecasting stuff, the radar,
00:53:25.272 --> 00:53:25.980
and all the rest.
00:53:25.980 --> 00:53:28.440
Just say it's going
to rain tomorrow.
00:53:28.440 --> 00:53:30.942
You're going to be right
almost all the time.
00:53:30.942 --> 00:53:31.620
All right?
00:53:31.620 --> 00:53:34.350
And in fact, if you
try to do fancy stuff,
00:53:34.350 --> 00:53:37.220
you're probably going to
be wrong more of the time.
00:53:37.220 --> 00:53:37.800
All right.
00:53:37.800 --> 00:53:40.130
For example, in this
case, if you just
00:53:40.130 --> 00:53:44.510
say the person does not have the
disease, forget the lab test.
00:53:44.510 --> 00:53:47.551
Just come back with negative.
00:53:47.551 --> 00:53:48.550
How often are you right?
00:53:51.870 --> 00:53:54.190
90% of the time you're right.
00:53:54.190 --> 00:53:56.460
Much better than the test
you paid a lot of money for.
00:53:59.050 --> 00:53:59.850
I see.
00:53:59.850 --> 00:54:02.630
You've got to be careful
what you're looking for,
00:54:02.630 --> 00:54:06.500
how you measure the value
of a test or a prediction.
00:54:06.500 --> 00:54:08.140
Because presumably
the one you paid
00:54:08.140 --> 00:54:14.240
for is better, even though
accurate less of the time.
00:54:14.240 --> 00:54:16.004
Any questions about that?
00:54:20.350 --> 00:54:22.030
OK.
00:54:22.030 --> 00:54:23.530
So For the rest of
today we're going
00:54:23.530 --> 00:54:27.160
to do three more paradoxes.
00:54:27.160 --> 00:54:29.990
And in each case
they're going to expose
00:54:29.990 --> 00:54:33.220
a flaw in our intuition
about probability.
00:54:33.220 --> 00:54:34.832
But the good news
is in each case it's
00:54:34.832 --> 00:54:36.040
easy to get the right answer.
00:54:36.040 --> 00:54:39.640
Just stick with the math and
try not to think about it.
00:54:39.640 --> 00:54:44.090
Now the first example
is a game involving
00:54:44.090 --> 00:54:48.240
dice that's called carnival dice
that you can find in carnivals
00:54:48.240 --> 00:54:50.175
and you can also
find in casinos.
00:54:53.150 --> 00:54:57.050
It's a pretty popular
game, actually.
00:54:57.050 --> 00:54:59.775
So the way it works
is as follows.
00:55:07.310 --> 00:55:21.360
The player picks a number from
1 to 6-- we'll call it N--
00:55:21.360 --> 00:55:23.650
and then rolls three dice.
00:55:23.650 --> 00:55:26.006
And let's say they're fair
and mutually independent.
00:55:34.970 --> 00:55:36.470
We haven't talked
about independent.
00:55:36.470 --> 00:55:38.460
So they're fair dice.
00:55:38.460 --> 00:55:41.250
For now, normal
dice-- nothing fishy.
00:55:41.250 --> 00:55:50.780
And the player wins if and
only if the number he picked
00:55:50.780 --> 00:55:52.455
comes up on at least
one of the dice.
00:55:58.100 --> 00:55:59.830
So you either win
or you lose the game
00:55:59.830 --> 00:56:03.375
depending on if your lucky
number came up at least once.
00:56:05.960 --> 00:56:08.590
Now you've got three dice,
each of which has a 1
00:56:08.590 --> 00:56:10.740
in 6 chance of coming
up a winner for you.
00:56:13.420 --> 00:56:17.730
So how many people think
this is a fair game-- you
00:56:17.730 --> 00:56:22.730
got a 50-50 chance of
winning-- three dice, each 1/6
00:56:22.730 --> 00:56:24.290
chance of winning?
00:56:24.290 --> 00:56:27.540
Anybody think it's
not a fair game?
00:56:27.540 --> 00:56:28.420
A bunch of you.
00:56:28.420 --> 00:56:31.840
How many people think it
is a fair game-- 50-50?
00:56:31.840 --> 00:56:32.601
A few.
00:56:32.601 --> 00:56:33.100
All right.
00:56:33.100 --> 00:56:36.479
Well, let's figure it out.
00:56:36.479 --> 00:56:38.270
And instead of doing
the tree method, which
00:56:38.270 --> 00:56:39.728
we know we're
supposed to do, we're
00:56:39.728 --> 00:56:46.284
just going to wing it, which
is always seems easier to do.
00:56:51.650 --> 00:56:54.180
If you're in the Casino
you want to just wing it
00:56:54.180 --> 00:56:57.970
instead of taking your napkin
out and drawing a tree.
00:56:57.970 --> 00:57:07.740
So the claim, question mark, is
the probability you win in 1/2.
00:57:07.740 --> 00:57:15.250
And the proof, question
mark, is you let Ai
00:57:15.250 --> 00:57:25.220
be the event that the i-th die
comes up N. And i is 1 to 3
00:57:25.220 --> 00:57:26.867
here.
00:57:26.867 --> 00:57:27.700
So then you say, OK.
00:57:27.700 --> 00:57:33.300
The probability I win is
the probability of A1--
00:57:33.300 --> 00:57:38.010
I could win that
way-- or A2, or A3.
00:57:38.010 --> 00:57:41.810
All I need is one of the
die to come up my way.
00:57:41.810 --> 00:57:45.430
And that is the
probability of A1
00:57:45.430 --> 00:57:52.210
plus the probability of A2
plus the probability of A3.
00:57:52.210 --> 00:57:54.555
And each die wins for
me with probability 1/6.
00:57:59.345 --> 00:58:00.220
And that is then 1/2.
00:58:03.460 --> 00:58:09.706
So that's a proof that we
win with probability of 1/2.
00:58:09.706 --> 00:58:13.160
What do you think?
00:58:13.160 --> 00:58:14.387
Any problems with that proof?
00:58:14.387 --> 00:58:15.262
AUDIENCE: [INAUDIBLE]
00:58:20.489 --> 00:58:22.030
PROFESSOR: Well
that's a great point.
00:58:22.030 --> 00:58:22.850
Yeah.
00:58:22.850 --> 00:58:25.950
So if I extended this
nice proof technique
00:58:25.950 --> 00:58:29.110
I couldn't have probability of
7/6 of winning with seven die.
00:58:29.110 --> 00:58:30.250
Yeah?
00:58:30.250 --> 00:58:31.125
AUDIENCE: [INAUDIBLE]
00:58:34.874 --> 00:58:35.540
PROFESSOR: Yeah.
00:58:35.540 --> 00:58:36.990
You're very close.
00:58:36.990 --> 00:58:41.062
I didn't technically
assume that.
00:58:41.062 --> 00:58:43.970
AUDIENCE: [INAUDIBLE]
00:58:43.970 --> 00:58:45.450
PROFESSOR: They could double up.
00:58:45.450 --> 00:58:46.580
Yeah.
00:58:46.580 --> 00:58:50.040
There's no intersection
in the events.
00:58:50.040 --> 00:58:53.220
In fact, there is
intersection because there's
00:58:53.220 --> 00:58:56.740
a chance I rolled
all six-- all Ns.
00:58:56.740 --> 00:58:57.680
Say N is 6.
00:58:57.680 --> 00:58:59.810
I could roll all sixes
and then each of these
00:58:59.810 --> 00:59:00.990
would be a winner.
00:59:00.990 --> 00:59:03.604
But I don't get to
count them separately.
00:59:03.604 --> 00:59:05.020
Then I only win
once in that case.
00:59:05.020 --> 00:59:07.290
In other words, all of
these could turned on
00:59:07.290 --> 00:59:08.060
at the same time.
00:59:08.060 --> 00:59:09.268
There's an intersection here.
00:59:09.268 --> 00:59:13.570
So this rule does not hold.
00:59:13.570 --> 00:59:16.290
I need the Ai to be
disjoined for this
00:59:16.290 --> 00:59:20.380
to be true-- the
events to be disjoined.
00:59:20.380 --> 00:59:22.110
And they're not
disjoined because there's
00:59:22.110 --> 00:59:25.040
a sample point were
two or more of the die
00:59:25.040 --> 00:59:28.470
could come up the same
being a winner, which
00:59:28.470 --> 00:59:33.560
means the same sample
point, namely all die are N,
00:59:33.560 --> 00:59:34.960
comes up in each of these three.
00:59:34.960 --> 00:59:36.530
So they're not disjoined.
00:59:36.530 --> 00:59:40.690
Now what's the principal
you used two weeks ago when
00:59:40.690 --> 00:59:46.560
you did cardinality of a set--
cardinality of a union of sets?
00:59:46.560 --> 00:59:48.480
Inclusion, exclusion.
00:59:48.480 --> 00:59:50.700
And the same thing
needs to be done here.
00:59:55.870 --> 00:59:57.364
So let's do that.
00:59:57.364 --> 00:59:59.405
And then we'll figure out
the actual probability.
01:00:03.450 --> 01:00:08.130
So this is a fact based on the
inclusion, exclusion principle.
01:00:08.130 --> 01:00:13.510
The probability of A1,
union A2, union A3,
01:00:13.510 --> 01:00:17.680
is just what you think it would
be from inclusion, exclusion.
01:00:17.680 --> 01:00:19.610
It's a probability of
A1 plus a probability
01:00:19.610 --> 01:00:26.550
of A2 plus the
probability of A3 minus
01:00:26.550 --> 01:00:27.720
the pairwise intersections.
01:00:33.700 --> 01:00:39.770
A1 intersect A3 minus
probability of A2 intersect A3.
01:00:42.310 --> 01:00:45.860
And is there anything else?
01:00:45.860 --> 01:00:48.300
Plus, the probably of
all of them matching.
01:00:54.860 --> 01:00:55.360
OK.
01:00:55.360 --> 01:00:59.070
So the proof is
really the same proof
01:00:59.070 --> 01:01:01.676
you use for inclusion,
exclusion with sets.
01:01:01.676 --> 01:01:03.800
The only difference is that
in a probability space,
01:01:03.800 --> 01:01:05.710
we have weights on the elements.
01:01:05.710 --> 01:01:09.190
And the weight corresponds
to the probability.
01:01:09.190 --> 01:01:14.510
So in fact, if you were drawing
the sample space, say here's A1
01:01:14.510 --> 01:01:19.790
and here's A2, and here's A3.
01:01:19.790 --> 01:01:24.860
Well, you need to add the
probabilities here, here,
01:01:24.860 --> 01:01:25.770
and here.
01:01:25.770 --> 01:01:30.180
Then you subtract off the double
counting from here, from here,
01:01:30.180 --> 01:01:30.790
and from here.
01:01:30.790 --> 01:01:33.560
And then you add back again
what you subtracted off
01:01:33.560 --> 01:01:36.460
too much there.
01:01:36.460 --> 01:01:39.430
Same proof, it's just your
have weights on the elements
01:01:39.430 --> 01:01:41.940
of probabilities.
01:01:41.940 --> 01:01:42.440
All right.
01:01:42.440 --> 01:01:45.280
So let's figure out
the right probability.
01:01:45.280 --> 01:01:50.550
That's 1/6, 1/6, 1/6.
01:01:50.550 --> 01:01:53.700
What's the probability
of the first two die
01:01:53.700 --> 01:01:54.980
matching-- both of them?
01:02:00.360 --> 01:02:02.640
1/36.
01:02:02.640 --> 01:02:05.110
We'll talk more about
why that is next time.
01:02:05.110 --> 01:02:09.600
But there's a 6 for A1 then
given that 1/6 for the second
01:02:09.600 --> 01:02:10.260
die matching.
01:02:10.260 --> 01:02:15.800
So it's 1/6 times
1/6 minus the 1/36.
01:02:15.800 --> 01:02:22.010
1/36, the chance that all three
match is 1/216 or 6 cubed.
01:02:22.010 --> 01:02:29.520
So when you add all that up you
get the 0.421 and some more.
01:02:29.520 --> 01:02:31.060
So the chance of
winning this game
01:02:31.060 --> 01:02:36.280
is 41% which makes it a
worst game in the casino.
01:02:36.280 --> 01:02:38.560
It is hard to find a
worse game than this.
01:02:38.560 --> 01:02:39.580
Roulette, much better.
01:02:39.580 --> 01:02:42.960
We'll study Roulette in the
last lecture-- much better game.
01:02:42.960 --> 01:02:45.990
And even that's a
terrible game to play.
01:02:45.990 --> 01:02:47.700
So it looks like an easy game.
01:02:47.700 --> 01:02:49.600
There's a quick proof
that it's 50-50.
01:02:49.600 --> 01:02:54.250
But it's horrible odds
against the house.
01:02:54.250 --> 01:02:57.660
Now, this is a nice
example because it
01:02:57.660 --> 01:03:01.490
shows how a rule you had for
computing the cardinality
01:03:01.490 --> 01:03:05.100
of a set gives you
the probability.
01:03:05.100 --> 01:03:05.620
All right.
01:03:05.620 --> 01:03:10.210
In fact, all the set laws you
learned a couple weeks ago
01:03:10.210 --> 01:03:12.969
work for probability
spaces the same way.
01:03:12.969 --> 01:03:14.760
And there were several
of those in homework
01:03:14.760 --> 01:03:18.400
that you just had
the last problem set.
01:03:18.400 --> 01:03:19.960
Any questions about that?
01:03:25.660 --> 01:03:27.090
OK.
01:03:27.090 --> 01:03:31.800
Now in addition, all those
set laws you did also
01:03:31.800 --> 01:03:35.070
work for conditional
probabilities.
01:03:35.070 --> 01:03:38.410
For example, this is true.
01:03:38.410 --> 01:03:44.690
The probability of A union B
given C-- whoops-- given C,
01:03:44.690 --> 01:03:50.860
is the probability of A given
C plus the probability of B
01:03:50.860 --> 01:03:57.030
given C minus the intersection,
A intersect B given
01:03:57.030 --> 01:04:02.180
C. In other words, take any
probability rule you have
01:04:02.180 --> 01:04:06.580
and condition everything on an
event, C, and it still works.
01:04:09.310 --> 01:04:11.340
And the proof is not hard.
01:04:11.340 --> 01:04:13.580
You can go through
each individual law
01:04:13.580 --> 01:04:17.550
but it all comes out to be fine.
01:04:17.550 --> 01:04:19.680
All right.
01:04:19.680 --> 01:04:21.770
You have to be a little
careful though because you
01:04:21.770 --> 01:04:25.030
got to remember which
side you're doing,
01:04:25.030 --> 01:04:27.580
which what you're putting on
either side of the bar here.
01:04:27.580 --> 01:04:29.830
For example, what
about this one?
01:04:29.830 --> 01:04:30.540
Is this true?
01:04:33.080 --> 01:04:34.900
Claim.
01:04:34.900 --> 01:04:37.400
Let's take-- say C
and D are disjoined.
01:04:42.900 --> 01:04:45.690
Is this true?
01:04:45.690 --> 01:04:49.870
Then the probability
of A conditioned
01:04:49.870 --> 01:04:56.450
on C union D. So given
that either C or D is true,
01:04:56.450 --> 01:05:00.810
does that equal the
probability of A given C
01:05:00.810 --> 01:05:04.420
plus probability of A given D?
01:05:07.140 --> 01:05:11.150
We know that if I swapped
all these, it's true.
01:05:11.150 --> 01:05:14.362
The probability of C union
D when C and D are disjoined
01:05:14.362 --> 01:05:16.820
is the probability that C given
A plus the probability of D
01:05:16.820 --> 01:05:19.030
given A. That I just claimed.
01:05:19.030 --> 01:05:20.070
And what about this way?
01:05:20.070 --> 01:05:21.685
Can I swap things around?
01:05:24.580 --> 01:05:26.306
Yeah?
01:05:26.306 --> 01:05:32.710
AUDIENCE: [INAUDIBLE]
would C union D be 0?
01:05:32.710 --> 01:05:36.350
PROFESSOR: If C and
D are disjoined,
01:05:36.350 --> 01:05:43.050
C union D would just be C union
D. But you're not a good point.
01:05:43.050 --> 01:05:44.540
What if C and D are disjoined?
01:05:44.540 --> 01:05:45.740
That's a good example.
01:05:45.740 --> 01:05:46.406
Let's draw that.
01:05:52.726 --> 01:05:53.726
Let's look at that case.
01:05:57.640 --> 01:06:00.540
So we've got a
sample space here.
01:06:00.540 --> 01:06:05.560
And you've got C
here and D here.
01:06:05.560 --> 01:06:09.745
And just for fun, let's make A
be here-- include all of them.
01:06:13.140 --> 01:06:18.680
What's the probability-- is
this going to do what I want?
01:06:18.680 --> 01:06:19.230
Yeah.
01:06:19.230 --> 01:06:21.504
What's the probability
of A given C?
01:06:25.440 --> 01:06:26.330
1.
01:06:26.330 --> 01:06:30.300
If I'm in C I'm in A.
A is everything here.
01:06:30.300 --> 01:06:32.970
So the probability
of A given C is one.
01:06:32.970 --> 01:06:36.255
What's the probability
of A given D?
01:06:36.255 --> 01:06:39.220
1.
01:06:39.220 --> 01:06:39.720
All right.
01:06:39.720 --> 01:06:41.277
Well, this is a
problem because I
01:06:41.277 --> 01:06:43.860
can't have the probability ot--
what's the probably of A given
01:06:43.860 --> 01:06:46.890
C union D?
01:06:46.890 --> 01:06:47.790
Well, it can't be 2.
01:06:47.790 --> 01:06:48.950
Right?
01:06:48.950 --> 01:06:50.550
It's 1.
01:06:50.550 --> 01:06:51.550
They are not equal.
01:06:54.120 --> 01:06:58.660
So you cannot do those set
rules on the right side
01:06:58.660 --> 01:07:01.020
of the conditioning bar.
01:07:01.020 --> 01:07:04.080
You can do them on the
left, not on the right.
01:07:04.080 --> 01:07:04.580
All right.
01:07:04.580 --> 01:07:05.413
So this is not true.
01:07:13.320 --> 01:07:14.391
Now nobody would do this.
01:07:14.391 --> 01:07:14.890
Right?
01:07:14.890 --> 01:07:18.140
I mean, the probability of-- not
that it's-- see this example?
01:07:18.140 --> 01:07:21.520
This you just would never
make this mistake again seeing
01:07:21.520 --> 01:07:23.450
that example.
01:07:23.450 --> 01:07:25.420
Everybody understand
the example,
01:07:25.420 --> 01:07:27.170
how it's clearly
not always the case
01:07:27.170 --> 01:07:28.770
that probability
of A given C union
01:07:28.770 --> 01:07:33.015
D is a probability of A given C
plus probability of A given D?
01:07:33.015 --> 01:07:35.390
Because now I'm going to show
you an example where you're
01:07:35.390 --> 01:07:38.250
going to swear it's true.
01:07:38.250 --> 01:07:39.372
All right?
01:07:39.372 --> 01:07:40.705
And this is a real life example.
01:07:43.660 --> 01:07:47.270
Many years ago now there was
a sex discrimination suit
01:07:47.270 --> 01:07:49.320
at Berkeley.
01:07:49.320 --> 01:07:52.170
There was a female professor
in the math department.
01:07:52.170 --> 01:07:54.500
And she was denied tenure.
01:07:54.500 --> 01:07:57.180
And she filed a lawsuit
against Berkeley
01:07:57.180 --> 01:07:59.680
alleging sex discrimination.
01:07:59.680 --> 01:08:02.390
Said she wasn't tenured
because she's a woman.
01:08:02.390 --> 01:08:04.661
Now, unfortunately
sex discrimination
01:08:04.661 --> 01:08:06.035
is a problem in
math departments.
01:08:06.035 --> 01:08:09.900
It's historically
been a difficult area.
01:08:09.900 --> 01:08:11.340
But it's always hard to prove.
01:08:11.340 --> 01:08:12.840
It's a nebulous kind of thing.
01:08:12.840 --> 01:08:14.590
They don't say, hey,
you can't have tenure
01:08:14.590 --> 01:08:15.810
because you're a woman.
01:08:15.810 --> 01:08:19.550
They'd get sued and
get killed for that.
01:08:19.550 --> 01:08:24.020
So she had to get some
mat to back her up.
01:08:24.020 --> 01:08:27.569
So what she did is she looked
into Berkeley's practices
01:08:27.569 --> 01:08:31.000
and she found that in
all 22 departments,
01:08:31.000 --> 01:08:34.040
every single department,
the percentage
01:08:34.040 --> 01:08:38.430
of male PhD applicants
that were accepted
01:08:38.430 --> 01:08:43.950
was higher than the percentage
of female PhD applicants
01:08:43.950 --> 01:08:46.069
that were accepted.
01:08:46.069 --> 01:08:49.069
Now you could understand some of
the departments accepting more
01:08:49.069 --> 01:08:50.809
male PhDs than female PhDs.
01:08:50.809 --> 01:08:53.430
But all 22?
01:08:53.430 --> 01:08:54.740
What are the odds of that?
01:08:54.740 --> 01:08:56.450
I mean, so the
immediate conclusion
01:08:56.450 --> 01:09:00.220
is, well, that's clearly there's
sex discrimination going on
01:09:00.220 --> 01:09:01.500
at Berkeley.
01:09:01.500 --> 01:09:03.399
OK?
01:09:03.399 --> 01:09:06.260
Well Berkeley took a look at
that and said, nothing good.
01:09:06.260 --> 01:09:08.130
That doesn't look good for them.
01:09:08.130 --> 01:09:13.140
But they did their own
study of PhD applicants.
01:09:13.140 --> 01:09:16.930
And they said that if the
university as a whole--
01:09:16.930 --> 01:09:21.040
look at the University as a
whole, actually, the women,
01:09:21.040 --> 01:09:25.029
the females have a higher
acceptance rate for the PhD
01:09:25.029 --> 01:09:27.609
Program than the men.
01:09:27.609 --> 01:09:28.109
So look.
01:09:28.109 --> 01:09:29.840
Berkeley said, we're
accepting more women
01:09:29.840 --> 01:09:32.540
than men percentage-wise.
01:09:32.540 --> 01:09:35.620
So how could we be
discriminating against women?
01:09:35.620 --> 01:09:38.670
And this is where the same
argument the female faculty
01:09:38.670 --> 01:09:41.010
member's making, But they're
saying as a university
01:09:41.010 --> 01:09:44.160
as a whole, when you add
up all 22 departments.
01:09:44.160 --> 01:09:45.410
Well, that sounds pretty good.
01:09:45.410 --> 01:09:47.790
How could they be
discriminating?
01:09:47.790 --> 01:09:48.290
OK.
01:09:48.290 --> 01:09:51.790
So the question for you
guys, is it possible that
01:09:51.790 --> 01:09:54.790
both sides we're
telling the truth,
01:09:54.790 --> 01:09:57.580
that in every single department
the women have a lower
01:09:57.580 --> 01:10:02.010
acceptance rate than men, but
on the university as a whole
01:10:02.010 --> 01:10:04.850
the women are higher percentage?
01:10:04.850 --> 01:10:12.450
It sounds like it's-- and just
to avoid any confusion here,
01:10:12.450 --> 01:10:17.330
people only apply to one
department and they're only one
01:10:17.330 --> 01:10:18.490
sex.
01:10:18.490 --> 01:10:21.916
So you can't--
Carroll didn't apply.
01:10:21.916 --> 01:10:25.620
[LAUGHTER]
01:10:25.620 --> 01:10:28.210
How many people think that one
of the sides, actually, when
01:10:28.210 --> 01:10:30.040
they look at the
studies was wrong,
01:10:30.040 --> 01:10:33.070
that they're contradictory?
01:10:33.070 --> 01:10:34.270
Nobody?
01:10:34.270 --> 01:10:35.852
You've been in 6
over 2 too long.
01:10:35.852 --> 01:10:37.310
How many people
think it's possible
01:10:37.310 --> 01:10:40.310
that both sides were right?
01:10:40.310 --> 01:10:40.941
Yeah.
01:10:40.941 --> 01:10:41.440
All right.
01:10:41.440 --> 01:10:44.410
So let's see how this works.
01:10:50.709 --> 01:10:52.500
And to make it simple
I'm going to get down
01:10:52.500 --> 01:10:56.500
to just two departments rather
than try to do data for all 22.
01:10:56.500 --> 01:10:59.706
And I'm going to do not the
actual data but something
01:10:59.706 --> 01:11:01.122
that's represents
what's going on.
01:11:05.931 --> 01:11:06.430
OK.
01:11:06.430 --> 01:11:08.388
So we're going to look
at the following events.
01:11:12.210 --> 01:11:19.340
A is the event that the
applicant is admitted.
01:11:25.930 --> 01:11:30.960
FCS is the event
that the applicant
01:11:30.960 --> 01:11:37.645
is female and applying to CS.
01:11:40.420 --> 01:11:43.770
FEE is the event
that the applicant
01:11:43.770 --> 01:11:46.815
is female and applying to EE.
01:11:49.540 --> 01:11:59.140
MCS is the event the
applicant is a male and CS.
01:11:59.140 --> 01:12:06.150
And then finally we have MEE
is the event the applicant is
01:12:06.150 --> 01:12:08.194
male and in EE.
01:12:08.194 --> 01:12:10.110
So we're just going to
look at two departments
01:12:10.110 --> 01:12:13.950
here and try to figure
out if it can happen
01:12:13.950 --> 01:12:16.760
that in both departments
the women are worse off
01:12:16.760 --> 01:12:19.148
but if you take the
union they're better off.
01:12:25.840 --> 01:12:30.070
So the female professor's
argument effectively
01:12:30.070 --> 01:12:34.030
is, the probability of being
admitted given that you're
01:12:34.030 --> 01:12:39.730
a female in CS is less than the
probability of being admitted
01:12:39.730 --> 01:12:43.080
given that you're a male at CS.
01:12:43.080 --> 01:12:46.820
And same thing in EE.
01:12:46.820 --> 01:12:50.210
Probability of being admitted
in EE if you're a female
01:12:50.210 --> 01:12:51.465
is less than if you're a male.
01:12:56.290 --> 01:12:57.680
OK?
01:12:57.680 --> 01:13:04.220
Now Berkeley is saying
it's sort of the reverse.
01:13:04.220 --> 01:13:07.740
The probability
that you're admitted
01:13:07.740 --> 01:13:13.260
given that you're a female
in either department
01:13:13.260 --> 01:13:17.530
is bigger than the probability
of being admitted if you're
01:13:17.530 --> 01:13:19.625
a male in either department.
01:13:24.720 --> 01:13:25.220
OK.
01:13:25.220 --> 01:13:28.180
So we've now expressed
their arguments
01:13:28.180 --> 01:13:33.310
as conditional
probabilities Any questions?
01:13:33.310 --> 01:13:37.045
Can you sort of see why
this seems contradictory?
01:13:39.610 --> 01:13:41.800
Not plus, union.
01:13:41.800 --> 01:13:45.990
Because this is sort of
like-- these are just joined.
01:13:45.990 --> 01:13:50.220
This is the sum of those.
01:13:50.220 --> 01:13:54.680
And this is sort of
the sum of those.
01:13:54.680 --> 01:13:58.900
And yet the inequality changed.
01:13:58.900 --> 01:13:59.400
All right.
01:13:59.400 --> 01:14:03.150
In fact, this is the logic
that we've just debunked over
01:14:03.150 --> 01:14:06.306
there-- exactly that claim.
01:14:06.306 --> 01:14:09.610
In fact, these are
not equal as the sum.
01:14:13.230 --> 01:14:14.270
So let's do an example.
01:14:18.370 --> 01:14:23.107
Say that-- let's
do it over here.
01:14:23.107 --> 01:14:24.690
I'll put the real
values in over here.
01:14:24.690 --> 01:14:28.170
Say that for women in
computer science, 0 out of 1
01:14:28.170 --> 01:14:32.040
were admitted compared to
the men, were 50 out of 100
01:14:32.040 --> 01:14:34.800
were admitted.
01:14:34.800 --> 01:14:37.680
And then in EE, 70
out of 100 women
01:14:37.680 --> 01:14:43.850
were admitted compared to the
men, which had 1 out of 1.
01:14:43.850 --> 01:14:44.350
All right?
01:14:44.350 --> 01:14:47.450
So as ratios, 70%
is less than 100%.
01:14:47.450 --> 01:14:49.620
0% is less than 50.
01:14:49.620 --> 01:14:51.620
Now if I look at the two
departments is a whole,
01:14:51.620 --> 01:15:00.830
I get 70 over 101 is in fact
bigger than 51 over 101.
01:15:00.830 --> 01:15:01.330
All right?
01:15:01.330 --> 01:15:02.790
And so as a whole
women are a lot more
01:15:02.790 --> 01:15:04.590
likely to be admitted even
though in each department
01:15:04.590 --> 01:15:06.048
they're less likely
to be admitted.
01:15:08.020 --> 01:15:10.800
OK?
01:15:10.800 --> 01:15:13.700
So what went wrong with
the intuition, which
01:15:13.700 --> 01:15:16.170
you didn't fall victim
to, but people often
01:15:16.170 --> 01:15:21.280
do, that it shouldn't have
been possible given that?
01:15:21.280 --> 01:15:22.710
What's going on
here that make it
01:15:22.710 --> 01:15:25.659
so that it's not
a less than when
01:15:25.659 --> 01:15:27.367
you look at the union
of the departments?
01:15:33.440 --> 01:15:34.066
Yeah?
01:15:34.066 --> 01:15:36.190
AUDIENCE: [INAUDIBLE]
they're weighted differently?
01:15:36.190 --> 01:15:36.856
PROFESSOR: Yeah.
01:15:36.856 --> 01:15:39.497
They're weighted
very differently.
01:15:39.497 --> 01:15:41.530
You got huge waves here.
01:15:41.530 --> 01:15:42.030
Right?
01:15:42.030 --> 01:15:45.740
So if I look at the average
of the percentages here,
01:15:45.740 --> 01:15:50.960
well it's 35% for the women
versus 75% for the men.
01:15:50.960 --> 01:15:53.860
So the average of the percentage
is just what you'd think.
01:15:53.860 --> 01:15:56.280
35 is less than 75.
01:15:56.280 --> 01:15:59.910
But I've got huge weightings
on these guys, which changes
01:15:59.910 --> 01:16:03.180
the numbers quite dramatically.
01:16:03.180 --> 01:16:04.776
So it all depends
how you count it.
01:16:08.432 --> 01:16:10.390
Actually, who do you
think had a better-- Yeah.
01:16:10.390 --> 01:16:11.226
Go ahead.
01:16:11.226 --> 01:16:12.120
AUDIENCE: [INAUDIBLE]
01:16:12.120 --> 01:16:13.411
PROFESSOR: Who won the lawsuit?
01:16:13.411 --> 01:16:15.910
Actually, the woman
won the lawsuit.
01:16:15.910 --> 01:16:18.179
And which argument
would you buy now?
01:16:18.179 --> 01:16:19.220
You've got two arguments.
01:16:19.220 --> 01:16:23.120
Which one would you
believe if either?
01:16:23.120 --> 01:16:24.250
Which one?
01:16:24.250 --> 01:16:28.820
I mean, now if I look
at exactly this data
01:16:28.820 --> 01:16:32.160
I might side-- I might
side with Berkeley
01:16:32.160 --> 01:16:34.400
looking at these numbers.
01:16:34.400 --> 01:16:38.190
Then again, when you think about
all 22 departments and the fact
01:16:38.190 --> 01:16:40.940
they weren't this
lopsided, not so good.
01:16:40.940 --> 01:16:43.407
So in the end Berkeley lost.
01:16:43.407 --> 01:16:45.240
I'm going to see another
example in a minute
01:16:45.240 --> 01:16:47.406
where it's even more clear
which side to believe in.
01:16:47.406 --> 01:16:49.330
But it really depends
on the numbers
01:16:49.330 --> 01:16:51.460
as to which one you
might, if you had to vote,
01:16:51.460 --> 01:16:54.070
which way you'd vote.
01:16:54.070 --> 01:16:56.390
Here's another example.
01:16:56.390 --> 01:17:00.820
This is from a newspaper article
on which airlines are best
01:17:00.820 --> 01:17:05.764
to fly because they have
the best on-time rates.
01:17:05.764 --> 01:17:10.300
And in this case they were
comparing American Airlines
01:17:10.300 --> 01:17:14.850
and America West,
looking at on-time rates.
01:17:14.850 --> 01:17:18.550
And here's the data they
showed for the two airlines.
01:17:18.550 --> 01:17:20.520
Here's American Airlines.
01:17:20.520 --> 01:17:23.510
Here's America West.
01:17:23.510 --> 01:17:31.070
And they took five cities,
LA, Phoenix, San Diego,
01:17:31.070 --> 01:17:33.105
San Francisco, and Seattle.
01:17:36.510 --> 01:17:41.350
And then you looked
at the number on time,
01:17:41.350 --> 01:17:47.349
the number of flights, and then
the rate, percentage on time.
01:17:47.349 --> 01:17:48.390
And then same thing here.
01:17:48.390 --> 01:17:57.300
Number on time, number
of flights, and the rate.
01:17:57.300 --> 01:17:59.800
So I'm just going to give
you the numbers here.
01:17:59.800 --> 01:18:14.100
So they had 500 out of 560 for
a rate of 89%, 220 over 230
01:18:14.100 --> 01:18:31.570
for 95, 210 over 230 for
92%, 500 over 600 for 83%,
01:18:31.570 --> 01:18:32.430
and then Seattle.
01:18:32.430 --> 01:18:33.580
They had a lot of flights.
01:18:33.580 --> 01:18:40.970
That's where they're-- we
have a hub of 2,200 for 86%.
01:18:40.970 --> 01:18:48.520
And if you added them all up,
they got 3,300 out of 3,820
01:18:48.520 --> 01:18:54.260
for 87% on time.
01:18:54.260 --> 01:18:56.260
Now the data for
American West looks
01:18:56.260 --> 01:18:58.150
something like the following.
01:18:58.150 --> 01:19:02.716
In LA it's 700 out
of 800 for 87%.
01:19:06.170 --> 01:19:07.470
they're based in Phoenix.
01:19:07.470 --> 01:19:08.860
They got a zillion
flights there.
01:19:08.860 --> 01:19:16.540
4,900 out of 5,300 for 92%.
01:19:16.540 --> 01:19:31.860
And 400 over 450 for 89%, 320,
over 450, 71%, 200 over 260
01:19:31.860 --> 01:19:34.290
for 77%.
01:19:34.290 --> 01:19:35.730
And then you add all them up.
01:19:35.730 --> 01:19:45.110
And you've got 6,520
over 7,260 for 90%.
01:19:45.110 --> 01:19:48.530
So the newspaper concluded
and literally said
01:19:48.530 --> 01:19:50.510
that American West
is the better airline
01:19:50.510 --> 01:19:53.410
to fly because they're
on-time rate is much better.
01:19:53.410 --> 01:19:55.825
It's 90% versus 87%.
01:19:58.880 --> 01:20:00.420
What do you think?
01:20:00.420 --> 01:20:03.960
Which airline would you
fly looking at that data?
01:20:03.960 --> 01:20:04.835
AUDIENCE: [INAUDIBLE]
01:20:10.550 --> 01:20:13.890
PROFESSOR: I know
which one I'd fly.
01:20:13.890 --> 01:20:16.790
It looks like America
West is better.
01:20:16.790 --> 01:20:20.680
Every single city, American
Airlines is better.
01:20:23.410 --> 01:20:25.380
92 versus 89.
01:20:25.380 --> 01:20:26.890
Everywhere it's
better by a bunch.
01:20:26.890 --> 01:20:29.010
83 versus 71.
01:20:29.010 --> 01:20:31.330
86 versus 77.
01:20:31.330 --> 01:20:36.510
Every single city, American
Airlines is better.
01:20:36.510 --> 01:20:40.422
Yet, America West
is better overall.
01:20:40.422 --> 01:20:41.880
And that's what
the newspaper said.
01:20:41.880 --> 01:20:43.196
They went on this.
01:20:43.196 --> 01:20:44.987
But of course, no matter
where you're going
01:20:44.987 --> 01:20:46.695
you're better off with
American Airlines.
01:20:48.961 --> 01:20:49.460
All right?
01:20:49.460 --> 01:20:53.880
Now what happened here?
01:20:53.880 --> 01:20:55.270
The waiting.
01:20:55.270 --> 01:20:59.460
In fact, America West
flies out of Phoenix
01:20:59.460 --> 01:21:02.520
where the weather's great.
01:21:02.520 --> 01:21:05.740
So you get a higher on-time rate
when in a good-weather city.
01:21:05.740 --> 01:21:07.690
And they got most of
their flights there.
01:21:07.690 --> 01:21:09.940
American Airlines got a
lot of flights in Seattle
01:21:09.940 --> 01:21:14.390
where the weather sucks
and you're always delayed.
01:21:14.390 --> 01:21:14.890
All right?
01:21:14.890 --> 01:21:16.530
And so they look
worse on average
01:21:16.530 --> 01:21:19.065
because so many of their
flights are in a bad city
01:21:19.065 --> 01:21:22.750
and so many of America
West are in a good city.
01:21:22.750 --> 01:21:23.250
All right?
01:21:23.250 --> 01:21:24.950
So it makes America
West look better
01:21:24.950 --> 01:21:28.420
when in fact, in this case, it's
absolutely clear whose better.
01:21:28.420 --> 01:21:31.422
American Airlines is
better, every single city.
01:21:31.422 --> 01:21:32.030
All right.
01:21:32.030 --> 01:21:34.860
That's why Mark
Twain said, "There's
01:21:34.860 --> 01:21:39.700
three kinds of lies-- lies,
damned lies, and statistics."
01:21:39.700 --> 01:21:42.410
We'll see more
examples next time.