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PROFESSOR: All right.

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00:00:21,500 --> 00:00:26,070
Today I want to spend a few more
minutes on plotting, and

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00:00:26,070 --> 00:00:30,390
then return to a subject that
will occupy us for a couple of

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weeks, which is the use of

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randomness in solving problems.

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Don't save.

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All right, so let's
first look at one

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more example of plotting.

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It's simple.

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00:00:48,210 --> 00:00:50,620
It's so simple you'll find
it's not in your handout.

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So here it is to start with.

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PROFESSOR: All I'm doing here is
I wrote a little program to

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show the effect of compound
interest, nothing very

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sophisticated.

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We start with some principal and
an interest rate, and then

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we just apply it over
and over again.

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And then we're going to plot to
show what the principal has

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00:01:22,890 --> 00:01:25,260
become if we just keep
compounding the interest.

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So it is kind of what
you'd expect.

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Compound interest is
a nice formula.

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You can actually get rich
applying it, and we see this

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nice little graph.

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On the other hand, we can't
really tell what it is.

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And this is the sort of thing
that I see all too often,

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including my graduate
students produce it.

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They come to my office, they
show me a graph, and they

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start explaining it.

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And I usually refuse to look at
it if it looks like this.

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There is no point in ever, and I
mean ever, producing a graph

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that does not have a title
and labeled axes.

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And in particular, you have
to label the axes to

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say what they mean.

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Fortunately, it's easy
enough to do.

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And here, I've just done that.

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So I'm going to run the same
code to compute the interest,

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but I'm going to put a
title on the graph.

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You've seen this before,
I just want to

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remind you how it works.

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PyLab.title And then I'm
going to label the

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x-axis and the y-axis.

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And that gives me a much
more useful graph.

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Nothing magical here, it's
just a reminder that you

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00:03:04,860 --> 00:03:07,780
really need to do
these things.

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You'll notice here I've not only
told you that this is the

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years of compounding and that
this is the principal but I've

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measured it in dollars.

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Maybe I should have been even
more explicit and said, well,

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US dollars, whatever.

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One of the things I did want
to point out is you saw the

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two of these various icons that
will let you do things

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like zoom in on a graph
and save a graph.

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Here's this icon that I think
Professor Grimson mentioned,

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in fact, I know he did.

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It's a floppy disk, just in case
you've never seen one, I

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brought a floppy disk
to show you.

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This is one of the older
floppy disks.

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These were invented
in 1971 by IBM.

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They were originally 8 inches in
diameter and held all of 80

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kilobytes of data.

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And as you can see, unlike
later floppy disks, they

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actually flopped.

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Eventually, Apple and others
pioneered a non-floppy floppy

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disk, that was in the '80s.

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The interesting thing today is I
typically carry around a USB

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stick with me about that big
that holds roughly 400,000

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times more data than
this floppy.

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And so it's just quite
incredible how things have

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gone along.

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All right.

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I now want to return to what
will be the main theme for, as

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I said, a couple of weeks
which is randomness.

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And in order to talk about
randomness we have to talk

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about probability.

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And I know that Professor
Grimson started down that path

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just before spring break, but
if you're anything like me

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your mind kind of deteriorated
a little bit over spring

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break, and your head isn't
quite into things.

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And so, I'm just going to back
up a tiny bit and start over

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to get our heads into it, and
then fairly quickly move on to

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new things.

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So let's start by asking
a simple question.

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You can tell my head isn't
quite yet back to things

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00:05:21,890 --> 00:05:28,520
because I forgot that I needed
to begin by gathering chalk.

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I've now got it.

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00:05:33,760 --> 00:05:36,015
And we'll come over here and
take a look at some examples.

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All right.

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00:05:44,110 --> 00:05:49,710
So the first question I want
to ask is, suppose I take a

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00:05:49,710 --> 00:05:54,630
6-sided die, a fair one, and I
roll it 10 times, what's the

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probability of not getting a
single 1, out of that die?

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00:06:02,450 --> 00:06:07,000
Well, how do we go about
answering this?

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00:06:07,000 --> 00:06:12,390
Well, there is a wrong way to
do it, which is sort of the

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00:06:12,390 --> 00:06:16,970
obvious way, and many people
will start down this path.

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They'll say, well the
probability of

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rolling a 1 on the--

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00:06:20,280 --> 00:06:23,950
not rolling a 1 on the first
try is 1 over 6.

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Right?

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00:06:31,030 --> 00:06:32,090
That's true?

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00:06:32,090 --> 00:06:32,950
That's not true.

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00:06:32,950 --> 00:06:36,230
What's the probability of not
rolling a 1 the first time?

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00:06:36,230 --> 00:06:37,510
5 over 6.

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All right.

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00:06:43,810 --> 00:06:48,280
What's the probability of not
rolling a 1 on the second try?

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00:06:48,280 --> 00:06:50,430
5 over 6.

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Well, the wrong thing to do, of
course, would be to start

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adding them up.

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We say, well, OK, we'll
just add these up.

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Well, one way we can tell that's
wrong is if we add up

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10 of these, we get
more than 1.

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00:07:11,040 --> 00:07:15,910
Probabilities can never be
more than 1 as we'll see.

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00:07:15,910 --> 00:07:18,050
So let's now try and think
of the right way to

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look at this problem.

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00:07:21,500 --> 00:07:24,160
So you can think about it.

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00:07:24,160 --> 00:07:27,230
If we roll these--

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00:07:27,230 --> 00:07:32,840
a die 10 times, each time
I'll get a number.

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00:07:32,840 --> 00:07:38,230
So I might get a 3, and then
a 4, and then a 2.

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How many possible 10-digit
numbers are there?

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00:07:42,470 --> 00:07:44,870
On a 6-sided die, if
I roll it 10 times?

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How many?

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00:07:51,840 --> 00:07:53,736
AUDIENCE: 6 to the 10th?

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PROFESSOR: 6 to the 10th.

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00:07:55,640 --> 00:08:02,870
Exactly Just when we look at
binary numbers, if I take a

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00:08:02,870 --> 00:08:06,170
10-digit binary number, and ask
how many different numbers

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00:08:06,170 --> 00:08:08,420
can I represent in 10 binary
digits, it's going

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00:08:08,420 --> 00:08:10,930
to be 2 to the 10th.

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00:08:10,930 --> 00:08:13,460
Here we're base 6.

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00:08:13,460 --> 00:08:16,470
So it's going to be
6 to the 10th.

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Pretty big number.

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00:08:18,660 --> 00:08:23,840
Now I can say, how many
of those numbers

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don't contain a 1?

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All right.

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00:08:30,550 --> 00:08:33,400
So that's really the question
I'm now asking.

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How many of these don't
contain a 1?

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00:08:36,830 --> 00:08:43,600
So as we said, if I look at the
first roll the odds of not

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getting a one the first time
is 5 over 6 Now what's the

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00:08:50,600 --> 00:08:56,060
odds of not getting 1 the first
or the second time?

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00:08:56,060 --> 00:09:01,575
It's 5 over 6 times 5 over 6.

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00:09:07,080 --> 00:09:08,750
That makes sense?

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00:09:08,750 --> 00:09:13,660
Because these are independent
events.

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00:09:13,660 --> 00:09:16,490
And that's a key notion here.

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00:09:16,490 --> 00:09:20,490
I'm assuming that whether I get
a 1 on the second roll is

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00:09:20,490 --> 00:09:22,885
independent of whether I got
a 1 on the first roll.

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It should be true, assuming
my dice--

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00:09:30,200 --> 00:09:32,560
die is fair.

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Similarly, I can do this for
the third roll et cetera.

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So the probability of not
getting a 1 in 10 rolls is

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00:09:46,260 --> 00:09:51,595
going to be (5 over
6) to the 10th.

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00:09:58,770 --> 00:10:01,170
That makes sense?

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00:10:01,170 --> 00:10:03,980
If not, speak up, because things
are going to get more

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00:10:03,980 --> 00:10:05,230
complicated quickly.

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00:10:08,250 --> 00:10:09,680
All right.

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00:10:09,680 --> 00:10:12,115
So that's pretty simple.

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00:10:16,570 --> 00:10:17,100
You--

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00:10:17,100 --> 00:10:18,690
you all-- are you all
with me on that?

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Now, suppose I ask you
the inverse question.

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00:10:26,360 --> 00:10:30,240
What is the probability of
getting at least one 1 if I

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00:10:30,240 --> 00:10:31,490
roll the die 10 times?

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00:10:35,320 --> 00:10:39,130
So here I've given you how to
compute the probability of not

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00:10:39,130 --> 00:10:40,810
getting any 1's.

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00:10:40,810 --> 00:10:46,290
Suppose I asked you the
probability of at least one 1?

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00:10:46,290 --> 00:10:46,665
Yeah?

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00:10:46,665 --> 00:10:48,040
AUDIENCE: [INAUDIBLE]

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00:10:48,040 --> 00:10:50,040
1 minus not having a 1?

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00:10:50,040 --> 00:10:51,040
PROFESSOR: Exactly.

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00:10:51,040 --> 00:10:52,540
Thank you.

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00:10:52,540 --> 00:11:02,430
So that would be 1 minus because
we know that the

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00:11:02,430 --> 00:11:06,900
probability, the sum of all the
possible things that we

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00:11:06,900 --> 00:11:08,680
can do when we do a probability

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00:11:08,680 --> 00:11:10,620
always has to be 1.

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00:11:16,230 --> 00:11:19,434
It was a good effort.

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00:11:19,434 --> 00:11:20,840
That's it.

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00:11:20,840 --> 00:11:21,860
If you take--

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00:11:21,860 --> 00:11:24,500
if you want to get something
where everything is covered,

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00:11:24,500 --> 00:11:28,700
the probabilities always
have to sum to 1.

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00:11:28,700 --> 00:11:31,640
And so now, there are only
two possibilities here.

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00:11:31,640 --> 00:11:34,760
One possibility is I
don't get any 1's.

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00:11:34,760 --> 00:11:39,350
One possibility is I
get at least one 1.

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00:11:39,350 --> 00:11:44,550
So if I take all of the
possibilities, and I subtract

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00:11:44,550 --> 00:11:48,020
the possibilities of not getting
any 1's, the result

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00:11:48,020 --> 00:11:52,800
must be the probability of
getting at least one 1.

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00:11:52,800 --> 00:11:58,710
This is a very common trick in
computing probabilities.

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00:11:58,710 --> 00:12:02,160
Very often when I ask or
somebody says, what's the

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00:12:02,160 --> 00:12:04,550
probability of x?

190
00:12:04,550 --> 00:12:07,470
The simplest way to compute
it, is to compute the

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00:12:07,470 --> 00:12:11,210
probability of not x and
subtract it from 1.

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00:12:17,700 --> 00:12:17,980
OK.

193
00:12:17,980 --> 00:12:22,350
Again, heading down a wrong
track for this, one might have

194
00:12:22,350 --> 00:12:26,010
said, well all right, the
probability of getting a 1 on

195
00:12:26,010 --> 00:12:32,020
the first roll is 1 over 6.

196
00:12:32,020 --> 00:12:33,960
The probability of getting
a 1 on the second

197
00:12:33,960 --> 00:12:35,235
roll is 1 over 6.

198
00:12:35,235 --> 00:12:38,850
The probability of getting
a third roll is 1 over 6.

199
00:12:38,850 --> 00:12:40,640
I'll just add them up, and
that will give me the

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00:12:40,640 --> 00:12:44,260
probability of getting
at least one one.

201
00:12:44,260 --> 00:12:45,080
How do I--

202
00:12:45,080 --> 00:12:47,320
how can I be sure
that's wrong?

203
00:12:47,320 --> 00:12:50,840
Well when I'm done, I would
claim the probability is

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00:12:50,840 --> 00:12:53,510
something like that.

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00:12:53,510 --> 00:12:55,470
And we know that
can't be true.

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00:12:55,470 --> 00:12:58,290
Because a probability always
has to be less

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00:12:58,290 --> 00:13:00,890
than or equal to 1.

208
00:13:00,890 --> 00:13:03,820
So this is a good trick to keep
in mind, whenever you're

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00:13:03,820 --> 00:13:07,840
given a probability problem, try
and figure out whether you

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00:13:07,840 --> 00:13:10,890
have a good way to compute it
directly, or whether it's

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00:13:10,890 --> 00:13:17,680
simpler to compute the not of
the probability, and then

212
00:13:17,680 --> 00:13:20,710
subtract it from 1.

213
00:13:20,710 --> 00:13:25,840
Probability is really
a fun field.

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00:13:25,840 --> 00:13:28,910
It's interesting, it's history,
it's intimately

215
00:13:28,910 --> 00:13:31,680
connected with the history
of gambling.

216
00:13:31,680 --> 00:13:36,710
And, in fact, almost all of
early probability theory owes

217
00:13:36,710 --> 00:13:40,090
its existence to gamblers.

218
00:13:40,090 --> 00:13:44,550
People like Cardano, Pascal,
Fermat, Bernoulli, de Moivre,

219
00:13:44,550 --> 00:13:48,990
Laplace, all famous names you've
heard, were motivated

220
00:13:48,990 --> 00:13:51,760
by desire to understand
games of chance.

221
00:13:55,260 --> 00:13:57,980
Mostly, it started with dice.

222
00:13:57,980 --> 00:14:00,100
I've been talking about dice.

223
00:14:00,100 --> 00:14:03,960
And in fact, dice are probably
the human race's oldest

224
00:14:03,960 --> 00:14:05,210
gambling implement.

225
00:14:07,530 --> 00:14:10,600
They date at least,
archaeologically, to about 600

226
00:14:10,600 --> 00:14:15,960
BC, where a pair of dice was
found in Egyptian tombs,

227
00:14:15,960 --> 00:14:17,660
actually longer than that.

228
00:14:17,660 --> 00:14:21,110
Two millennia before the birth
of Christ, people found dice

229
00:14:21,110 --> 00:14:22,360
in Egyptian tombs.

230
00:14:24,580 --> 00:14:27,350
Typically, they were made
from animal bones, but

231
00:14:27,350 --> 00:14:28,960
that doesn't matter.

232
00:14:28,960 --> 00:14:32,230
Pascal's interest in it, and
Pascal is really considered

233
00:14:32,230 --> 00:14:35,840
the founder of probability
theory, came when a friend

234
00:14:35,840 --> 00:14:38,730
asked him to solve the following
problem which I want

235
00:14:38,730 --> 00:14:42,300
to work out with you.

236
00:14:42,300 --> 00:14:47,590
Is it profitable to bet that
given 24 rolls of a pair of

237
00:14:47,590 --> 00:14:51,535
fair dice, you would
roll a double 6?

238
00:14:54,220 --> 00:14:57,070
He actually had a friend who
was in the business of

239
00:14:57,070 --> 00:14:59,180
gambling, making these bets.

240
00:14:59,180 --> 00:15:07,290
So he said, you've got a pair of
dice, you roll it 24 times

241
00:15:07,290 --> 00:15:12,830
and ask the question, what is
the probability of getting

242
00:15:12,830 --> 00:15:16,220
what we call today "box cars",
in those days they

243
00:15:16,220 --> 00:15:19,540
just called two 6's.

244
00:15:19,540 --> 00:15:23,060
This was considered a really
hard problem in

245
00:15:23,060 --> 00:15:25,430
the mid-17th century.

246
00:15:25,430 --> 00:15:31,080
And in fact, Pascal and Fermat,
two pretty smart guys

247
00:15:31,080 --> 00:15:33,190
as it happens, debated this.

248
00:15:33,190 --> 00:15:35,540
They exchanged letters with each
other trying to figure

249
00:15:35,540 --> 00:15:37,490
out how to solve this problem.

250
00:15:37,490 --> 00:15:40,920
It shows how math has advanced
because, in fact, today, it's

251
00:15:40,920 --> 00:15:42,830
quite an easy problem.

252
00:15:42,830 --> 00:15:45,680
So let's work it through
and think how would

253
00:15:45,680 --> 00:15:50,070
we answer this question.

254
00:15:50,070 --> 00:16:00,560
So what's the probability of
rolling, of not rolling, a

255
00:16:00,560 --> 00:16:03,475
double 6 on the first try?

256
00:16:06,910 --> 00:16:10,650
Well, the probability of
not rolling a 6 on one

257
00:16:10,650 --> 00:16:12,000
die is a sixth --

258
00:16:12,000 --> 00:16:13,900
1 over 6.

259
00:16:13,900 --> 00:16:16,900
The probability of not rolling
a one with the next die is

260
00:16:16,900 --> 00:16:20,170
also 1 over 6.

261
00:16:20,170 --> 00:16:23,340
So the probability of not
getting a die in the first

262
00:16:23,340 --> 00:16:26,760
roll, first double 6's is--

263
00:16:29,750 --> 00:16:34,170
the probability of getting
a double 6 is 1/36.

264
00:16:34,170 --> 00:16:38,395
So the probability of not
getting a double 6 is 35/36.

265
00:16:41,800 --> 00:16:44,350
Right?

266
00:16:44,350 --> 00:16:48,880
So now we know that the
probability of not

267
00:16:48,880 --> 00:16:50,170
getting it is that.

268
00:16:52,690 --> 00:16:57,050
What's the probability of not
getting it 24 times in a row?

269
00:17:00,310 --> 00:17:01,560
It's that.

270
00:17:05,960 --> 00:17:12,000
Which is approximately
equal to 0.51.

271
00:17:12,000 --> 00:17:15,550
So you can see why the answer
was not obvious just by

272
00:17:15,550 --> 00:17:17,670
experience.

273
00:17:17,670 --> 00:17:24,960
But there is a slight edge in
betting that you will not get

274
00:17:24,960 --> 00:17:28,369
a double 6 in 24 times.

275
00:17:28,369 --> 00:17:30,580
Again, assuming you
have fair dice.

276
00:17:30,580 --> 00:17:34,220
As old as dice is, people have
built cheater's dice.

277
00:17:34,220 --> 00:17:37,450
The excavation of Pompeii, for
example, they discovered a

278
00:17:37,450 --> 00:17:41,050
pair of loaded dice, dice with
a little weight in it so one

279
00:17:41,050 --> 00:17:45,100
number would come up more
often than it should.

280
00:17:45,100 --> 00:17:47,210
And in fact, if you look at the
internet today, you will

281
00:17:47,210 --> 00:17:51,880
find many sites where you can,
let's see, the one I found

282
00:17:51,880 --> 00:17:55,290
this morning says quote, "Are
you on unusually unlucky when

283
00:17:55,290 --> 00:17:57,780
it comes to rolling dice?

284
00:17:57,780 --> 00:18:01,640
Investing in a pair of dice
that's more reliable might be

285
00:18:01,640 --> 00:18:05,770
just what you need." And then
it says, "Of course for

286
00:18:05,770 --> 00:18:10,530
amusement only." Yeah,
we believe that.

287
00:18:10,530 --> 00:18:12,420
All right.

288
00:18:12,420 --> 00:18:17,050
As much as I trust probability
theory, I don't trust my

289
00:18:17,050 --> 00:18:18,940
ability to use it.

290
00:18:18,940 --> 00:18:28,270
And so what I did is wrote a
little simulation to see if

291
00:18:28,270 --> 00:18:32,240
Pascal was right when
he did this.

292
00:18:32,240 --> 00:18:38,130
So I've got the first-- just
this little test roll, which

293
00:18:38,130 --> 00:18:42,205
rolls a dice number of times,
gets the result.

294
00:18:46,360 --> 00:18:50,310
Now, then I decided
to check Pascal.

295
00:18:50,310 --> 00:18:55,300
So I was going to run 100,000
trials, and keep track of the

296
00:18:55,300 --> 00:18:57,940
number of times it worked.

297
00:18:57,940 --> 00:19:00,880
So what you'll see I'm doing
here is for i in the range

298
00:19:00,880 --> 00:19:04,400
number of trials, and this is
the way we'll do a lot of

299
00:19:04,400 --> 00:19:06,760
these simulations.

300
00:19:06,760 --> 00:19:12,110
And in fact, as we deal with
probability, we'll be dealing

301
00:19:12,110 --> 00:19:19,580
a lot with the notion of
simulation, as you are doing

302
00:19:19,580 --> 00:19:22,950
in your current problem set.

303
00:19:22,950 --> 00:19:28,770
So for i in range number of
trials, for j in range 24,

304
00:19:28,770 --> 00:19:33,500
because that was Pascal's
friend's game, I'll roll the

305
00:19:33,500 --> 00:19:36,230
first die, I'll roll
the second die.

306
00:19:36,230 --> 00:19:39,710
If they're both 6's I'll
say, yes equals 1.

307
00:19:39,710 --> 00:19:47,190
And I'll break and then I'll
compute the probability of

308
00:19:47,190 --> 00:19:48,440
winning or losing.

309
00:19:50,800 --> 00:19:52,050
OK?

310
00:19:54,010 --> 00:19:57,258
So let's let it rip.

311
00:19:57,258 --> 00:19:58,970
So now let's let it rip.

312
00:20:11,150 --> 00:20:12,860
There it is.

313
00:20:12,860 --> 00:20:15,670
And we can see that it actually
comes out pretty

314
00:20:15,670 --> 00:20:20,980
close to what Pascal
predicted.

315
00:20:20,980 --> 00:20:24,560
Should we be surprised that it
didn't come out to exactly?

316
00:20:24,560 --> 00:20:27,030
Well let's see, is it exactly?

317
00:20:27,030 --> 00:20:30,330
What is 35/36 to the 24th?

318
00:20:46,120 --> 00:20:48,482
So that's the--

319
00:20:48,482 --> 00:20:52,630
well, to 17 digits of precision,
the exact answer.

320
00:20:52,630 --> 00:20:55,950
And you can see we came up with
something close to that.

321
00:20:55,950 --> 00:20:59,470
Not exactly that, and we
wouldn't expect to.

322
00:20:59,470 --> 00:21:01,990
Now I only did 100,000 trials.

323
00:21:01,990 --> 00:21:05,440
If I did a million trials,
I'd probably come up with

324
00:21:05,440 --> 00:21:09,410
something even closer, but if I
did 2 trials, who knows what

325
00:21:09,410 --> 00:21:11,244
I get-- come up with it, right?

326
00:21:11,244 --> 00:21:13,464
Could be--

327
00:21:13,464 --> 00:21:19,010
I could get 1, I could get lucky
both times, or unlucky.

328
00:21:19,010 --> 00:21:22,200
Later on, we'll talk more about
the question, how do we

329
00:21:22,200 --> 00:21:25,460
know how many trials to run?

330
00:21:25,460 --> 00:21:31,030
Now, the interesting thing is
I'm sure it took me less time

331
00:21:31,030 --> 00:21:33,040
to write this program
than it took Pascal

332
00:21:33,040 --> 00:21:34,820
to solve the problem.

333
00:21:34,820 --> 00:21:37,410
Now the truth is, I had several
hundred years of other

334
00:21:37,410 --> 00:21:39,690
people's work to build on.

335
00:21:39,690 --> 00:21:44,270
But in general, I think one of
the questions you'll find is,

336
00:21:44,270 --> 00:21:48,365
is it easier sometimes to write
a simulation, than it is

337
00:21:48,365 --> 00:21:53,230
to do the probabilities?

338
00:21:53,230 --> 00:21:57,550
What I often do in
practice is both.

339
00:21:57,550 --> 00:22:00,920
I'll scratch my head and figure
out how to figure out

340
00:22:00,920 --> 00:22:04,060
the answer analytically, and
then if it's easy, I'll write

341
00:22:04,060 --> 00:22:08,850
some code to simulate the
problem, and expect to get

342
00:22:08,850 --> 00:22:12,310
roughly the same answer, giving
me confidence I've done

343
00:22:12,310 --> 00:22:14,000
it correctly.

344
00:22:14,000 --> 00:22:16,400
On the other hand, if I've done
the simulation and it had

345
00:22:16,400 --> 00:22:20,980
come up with something totally
bogus, or totally different,

346
00:22:20,980 --> 00:22:23,150
then I would have had to work
hard to figure out which was

347
00:22:23,150 --> 00:22:27,400
right, the code or the math.

348
00:22:27,400 --> 00:22:29,770
Same sort of thing you saw when
you looked at the random

349
00:22:29,770 --> 00:22:33,340
walk, and the first time it was
done an answer showed up

350
00:22:33,340 --> 00:22:36,340
that was just wrong.

351
00:22:36,340 --> 00:22:40,730
But, you need to have some
intuition about a problem, so

352
00:22:40,730 --> 00:22:42,940
that you can look at
it and say, yeah,

353
00:22:42,940 --> 00:22:44,680
that's in the ballpark.

354
00:22:44,680 --> 00:22:48,790
And if it's not, it's
time to worry.

355
00:22:48,790 --> 00:22:51,770
This kind of simulation that
I've just done for the dice

356
00:22:51,770 --> 00:22:54,886
game is what's called a "Monte
Carlo simulation".

357
00:23:03,660 --> 00:23:11,850
It is the most popular kind of
simulation named after a

358
00:23:11,850 --> 00:23:21,940
Casino on the Riviera, in the
small principality of Monaco.

359
00:23:21,940 --> 00:23:24,420
This was back in the time when
it was hard to find a place

360
00:23:24,420 --> 00:23:26,840
you could gamble, and this
happened to be one of the

361
00:23:26,840 --> 00:23:30,180
places you could.

362
00:23:30,180 --> 00:23:35,865
The term was coined in 1949 by
Stanislaw Ulam and Nicholas

363
00:23:35,865 --> 00:23:42,110
Metropolis, two very well-known
mathematicians.

364
00:23:42,110 --> 00:23:46,020
Ulam, who later became famous
for designing the hydrogen

365
00:23:46,020 --> 00:23:53,300
bomb with Teller, invented the
method in 1946, and I'm going

366
00:23:53,300 --> 00:23:58,130
to quote from his description
of how he invented it.

367
00:23:58,130 --> 00:24:00,710
"The first thoughts and attempts
I made to practice

368
00:24:00,710 --> 00:24:04,180
the Monte Carlo method, were
suggested by a question which

369
00:24:04,180 --> 00:24:08,110
occurred to me in 1946, as I
was convalescing from an

370
00:24:08,110 --> 00:24:11,010
illness and playing
solitaires.

371
00:24:11,010 --> 00:24:13,740
The question was, what are the
chances that a canfield

372
00:24:13,740 --> 00:24:16,990
solitaire laid out with
52 cards will come out

373
00:24:16,990 --> 00:24:18,990
successfully?

374
00:24:18,990 --> 00:24:21,880
After spending a lot of time
trying to estimate them by

375
00:24:21,880 --> 00:24:25,310
pure combinatorial calculations,
I wondered

376
00:24:25,310 --> 00:24:28,800
whether a more practical method
than quote 'abstract

377
00:24:28,800 --> 00:24:33,210
thinking' end quote, might not
be to lay it out, say, 100

378
00:24:33,210 --> 00:24:36,520
times, and simply observe
and count the number

379
00:24:36,520 --> 00:24:39,500
of successful plays.

380
00:24:39,500 --> 00:24:42,790
This was already possible to
envision with the beginning of

381
00:24:42,790 --> 00:24:45,920
the new era of fast computers.

382
00:24:45,920 --> 00:24:48,780
And I immediately thought of
problems, as you would, I'm

383
00:24:48,780 --> 00:24:52,010
sure, immediately thought of
problems of neutron diffusion

384
00:24:52,010 --> 00:24:55,260
and other questions of
mathematical physics.

385
00:24:55,260 --> 00:24:59,040
And more generally, how to
change processes described by

386
00:24:59,040 --> 00:25:02,680
certain differential equations
into an equivalent form

387
00:25:02,680 --> 00:25:07,610
interpretable as a succession
of random operations.

388
00:25:07,610 --> 00:25:11,480
Later, I described the idea to
John von Neumann, and we began

389
00:25:11,480 --> 00:25:14,460
to plan actual calculations."

390
00:25:14,460 --> 00:25:17,880
So as early as 1946, people
were thinking about the

391
00:25:17,880 --> 00:25:22,820
question of moving away from
solving systems of equations,

392
00:25:22,820 --> 00:25:28,230
to using randomized techniques
to simulate things and try to

393
00:25:28,230 --> 00:25:32,600
find out what the actual
answer was that way.

394
00:25:32,600 --> 00:25:35,810
Now of course "fast"
is a relative term.

395
00:25:35,810 --> 00:25:39,820
Ulam was probably referring to
the ENIAC computer, which

396
00:25:39,820 --> 00:25:44,180
could perform about 10 to the
3 additions a second.

397
00:25:44,180 --> 00:25:47,110
Not very many, 1,000 operations
a second, and

398
00:25:47,110 --> 00:25:50,760
weighed approximately 25 tons.

399
00:25:50,760 --> 00:25:53,620
Now today's computers, by
comparison, perform 10 to the

400
00:25:53,620 --> 00:25:58,490
9th additions and weigh about
10 to the minus 3 tons.

401
00:25:58,490 --> 00:25:59,090
All right.

402
00:25:59,090 --> 00:26:02,390
This technique was used during
the Manhattan Project to

403
00:26:02,390 --> 00:26:04,220
predict what would
happen doing--

404
00:26:04,220 --> 00:26:07,050
during nuclear fission
and worked.

405
00:26:09,550 --> 00:26:15,950
Monte Carlo simulations are an
example of what's called

406
00:26:15,950 --> 00:26:17,200
"inferential statistics".

407
00:26:28,400 --> 00:26:30,930
In brief, and I'm going to be
brief because this is not a

408
00:26:30,930 --> 00:26:35,260
statistics, course, inferential
statistics is

409
00:26:35,260 --> 00:26:39,440
based upon one guiding
principle.

410
00:26:39,440 --> 00:26:58,200
And that principle is that a
random sample tends to exhibit

411
00:26:58,200 --> 00:27:09,500
the same properties
as the population

412
00:27:09,500 --> 00:27:10,750
from which it is drawn.

413
00:27:18,550 --> 00:27:22,110
So if I try and sample people,
say, for predicting an

414
00:27:22,110 --> 00:27:26,820
election, the notion is if I go
and I asked a 1,000 people

415
00:27:26,820 --> 00:27:30,710
at random in Massachusetts who
they're going to vote for, the

416
00:27:30,710 --> 00:27:34,570
average will be about the same
as if I looked at the whole

417
00:27:34,570 --> 00:27:35,820
population.

418
00:27:37,550 --> 00:27:41,940
So whenever we use a statistical
method like this,

419
00:27:41,940 --> 00:27:46,610
so for example, we assumed here,
is those 100,000 times I

420
00:27:46,610 --> 00:27:50,730
threw the pair of dice, that
that would be representative

421
00:27:50,730 --> 00:27:54,460
of all possible throws of the
dice, the infinite number of

422
00:27:54,460 --> 00:27:55,710
possible throws.

423
00:27:58,890 --> 00:28:04,210
One always has to ask the
question whether this is true,

424
00:28:04,210 --> 00:28:06,810
or whether one has a sampling
technique that is, for

425
00:28:06,810 --> 00:28:09,560
example, giving you
a biased sample.

426
00:28:09,560 --> 00:28:13,260
Little later in the term, we'll
talk about many ways in

427
00:28:13,260 --> 00:28:17,730
which you can get fooled here
and think you're doing a fair

428
00:28:17,730 --> 00:28:21,190
statistical analysis, and get
all the math right, and still

429
00:28:21,190 --> 00:28:25,190
come up with the wrong answer
because this assumption

430
00:28:25,190 --> 00:28:27,480
doesn't actually hold.

431
00:28:27,480 --> 00:28:28,240
All right.

432
00:28:28,240 --> 00:28:31,360
Let's think about it now in
terms of coins, a little

433
00:28:31,360 --> 00:28:34,590
simpler than dice, where you
can flip a coin and you get

434
00:28:34,590 --> 00:28:35,840
either a head or a tail.

435
00:28:38,250 --> 00:28:42,120
Suppose Harvey Dent, for
example, flipped a coin and it

436
00:28:42,120 --> 00:28:43,370
came up heads.

437
00:28:45,500 --> 00:28:48,980
Would you feel good inferring
from that that the next time

438
00:28:48,980 --> 00:28:51,100
he flipped a coin it would
also come up heads?

439
00:28:54,510 --> 00:28:56,310
I wouldn't.

440
00:28:56,310 --> 00:28:58,150
Suppose he flipped it
heads and it came up

441
00:28:58,150 --> 00:29:00,920
heads twice, in a row.

442
00:29:00,920 --> 00:29:02,510
Would you feel comfortable
with the third

443
00:29:02,510 --> 00:29:05,410
flip would be a head?

444
00:29:05,410 --> 00:29:07,540
Probably not.

445
00:29:07,540 --> 00:29:11,870
But suppose he flipped it a 100
times in a row, and it was

446
00:29:11,870 --> 00:29:13,040
a head each time.

447
00:29:13,040 --> 00:29:15,590
What would you infer?

448
00:29:15,590 --> 00:29:19,820
I would infer that the coin
two-headed And, in fact, every

449
00:29:19,820 --> 00:29:22,530
time it was going to come up
heads, because it is so

450
00:29:22,530 --> 00:29:26,070
improbable that if it
was a fair coin--

451
00:29:26,070 --> 00:29:28,290
what's the probability of having
a 100 heads in a row

452
00:29:28,290 --> 00:29:29,540
with a fair coin?

453
00:29:32,170 --> 00:29:34,525
1 over what?

454
00:29:34,525 --> 00:29:37,129
AUDIENCE: 1 over 100--

455
00:29:37,129 --> 00:29:39,021
1 over 2 to the 100th.

456
00:29:39,021 --> 00:29:39,967
Right?

457
00:29:39,967 --> 00:29:42,600
PROFESSOR: A half the
first time times a

458
00:29:42,600 --> 00:29:43,830
half times a half.

459
00:29:43,830 --> 00:29:47,550
A huge number, a very small
number rather, right?

460
00:29:47,550 --> 00:29:51,370
So the probability and a fair
coin of getting hundred heads

461
00:29:51,370 --> 00:29:56,360
in a row is so low with just 100
flips, that I would begin

462
00:29:56,360 --> 00:30:00,910
to think that the coin
was not fair.

463
00:30:00,910 --> 00:30:02,940
All right.

464
00:30:02,940 --> 00:30:10,680
Suppose, however, I flipped it
100 times and I got 52 heads

465
00:30:10,680 --> 00:30:11,930
and 48 tails.

466
00:30:15,760 --> 00:30:18,700
Well, I wouldn't assume
anything from that.

467
00:30:18,700 --> 00:30:22,110
Would I assume that the next
time I flipped it a 100 times

468
00:30:22,110 --> 00:30:26,380
I'd get the same
52 to 48 ratio?

469
00:30:26,380 --> 00:30:27,670
Probably not, right?

470
00:30:27,670 --> 00:30:31,940
Your common sense tells
you you wouldn't.

471
00:30:31,940 --> 00:30:33,360
All right.

472
00:30:33,360 --> 00:30:36,010
Probably, it tells you, you
wouldn't even feel comfortable

473
00:30:36,010 --> 00:30:37,520
guessing that there would
be more heads than

474
00:30:37,520 --> 00:30:38,770
tails the next time.

475
00:30:42,860 --> 00:30:47,310
So when we think about these
things, we have to think about

476
00:30:47,310 --> 00:30:53,830
the number of tests and how
close the answer is to what

477
00:30:53,830 --> 00:30:57,920
you would get if you did
things at random.

478
00:30:57,920 --> 00:30:59,210
This is sort of comparing--

479
00:30:59,210 --> 00:31:01,830
this is technically called
comparing something to the

480
00:31:01,830 --> 00:31:03,980
null hypothesis.

481
00:31:03,980 --> 00:31:06,760
The null hypothesis is
what you would get

482
00:31:06,760 --> 00:31:09,110
with a random event.

483
00:31:09,110 --> 00:31:16,080
And when you do a simulation,
if you get something that is

484
00:31:16,080 --> 00:31:19,420
far from that, or when you
sample a population, you get

485
00:31:19,420 --> 00:31:22,950
something that's distant from
the null hypothesis, you can

486
00:31:22,950 --> 00:31:26,990
assume that maybe you're
seeing something real.

487
00:31:26,990 --> 00:31:27,250
All right.

488
00:31:27,250 --> 00:31:30,130
Let's look at this in a little
less abstract way.

489
00:31:40,860 --> 00:31:43,600
So let's go look at
some coin flips.

490
00:31:43,600 --> 00:31:47,590
So I wrote a simple
program, flip.

491
00:31:47,590 --> 00:31:54,130
Just flip the coin some number
of times and tells me what

492
00:31:54,130 --> 00:31:55,380
fraction came up heads.

493
00:32:00,260 --> 00:32:04,460
So we can run it, and
let's look at a--

494
00:32:04,460 --> 00:32:11,402
suppose I flip a 100,
I get 0.55.

495
00:32:11,402 --> 00:32:15,440
If I flip 10, I get 0.4.

496
00:32:15,440 --> 00:32:23,490
If I flip 10 again, I get 0.5.

497
00:32:23,490 --> 00:32:28,990
Now look at that, the same
thing twice in a row

498
00:32:28,990 --> 00:32:31,330
but now I get 0.2.

499
00:32:31,330 --> 00:32:36,330
So obviously, I shouldn't infer
too much from 10 flips

500
00:32:36,330 --> 00:32:39,030
and even from 100 where
I got 0.55.

501
00:32:39,030 --> 00:32:45,350
Let's see what happens if I
flip 100 again, 0.41, big

502
00:32:45,350 --> 00:32:47,870
difference.

503
00:32:47,870 --> 00:32:51,170
So this is suggesting that we
can't feel very good about

504
00:32:51,170 --> 00:32:53,610
what happens here.

505
00:32:53,610 --> 00:33:08,330
Now if I do the following, well
I'm feeling a little bit

506
00:33:08,330 --> 00:33:13,090
better about this, well
for one bad reason

507
00:33:13,090 --> 00:33:14,110
and one good reason.

508
00:33:14,110 --> 00:33:17,900
The bad reason is, I know the
answers 0.5, and these are

509
00:33:17,900 --> 00:33:20,805
both close to 0.5, so I
feel warm and fuzzy.

510
00:33:20,805 --> 00:33:21,900
But that's cheating.

511
00:33:21,900 --> 00:33:23,700
I wouldn't need to write
the simulation

512
00:33:23,700 --> 00:33:25,820
If I knew the answer.

513
00:33:25,820 --> 00:33:29,530
But mostly I feel good about it
because I'm getting kind of

514
00:33:29,530 --> 00:33:31,150
the same answer every time.

515
00:33:34,060 --> 00:33:37,050
OK, and that's important.

516
00:33:37,050 --> 00:33:42,540
The more I do, the more stable
it gets with the larger the

517
00:33:42,540 --> 00:33:45,070
number of trials.

518
00:33:45,070 --> 00:33:50,470
This is an example of what's
called "the law of large

519
00:33:50,470 --> 00:34:10,780
numbers", also known as
Bernoulli's Law, after one of

520
00:34:10,780 --> 00:34:13,210
the Bernoulli family of
mathematicians, and I can't

521
00:34:13,210 --> 00:34:15,889
for the life of me remember
which Bernoulli.

522
00:34:15,889 --> 00:34:18,290
There are a whole
bunch of them.

523
00:34:18,290 --> 00:34:22,199
Anyway the law states, and it's
important to understand

524
00:34:22,199 --> 00:34:25,270
this because again it underlies
the inferential

525
00:34:25,270 --> 00:34:38,940
statistics, that in repeated
independent tests, and it's

526
00:34:38,940 --> 00:34:44,600
important to note the word
"independent", each test has

527
00:34:44,600 --> 00:34:47,469
to be independent of
the earlier test.

528
00:34:47,469 --> 00:34:57,070
In this case, the tests are
flips of the coin with the

529
00:34:57,070 --> 00:35:11,910
same actual probability we'll
call it p, often used to

530
00:35:11,910 --> 00:35:22,030
represent probability, of an
outcome for each test, the

531
00:35:22,030 --> 00:35:41,350
chance that the fraction of
times that outcome occurs the

532
00:35:41,350 --> 00:35:56,380
outcome that with probability,
p, converges to p as number of

533
00:35:56,380 --> 00:35:57,700
trials goes to infinity.

534
00:36:09,300 --> 00:36:09,680
All right.

535
00:36:09,680 --> 00:36:14,240
So if I did an infinite number
of trials, the fraction of

536
00:36:14,240 --> 00:36:20,850
heads I would get in this case
would be exactly 0.5.

537
00:36:20,850 --> 00:36:24,370
Of course I can't do an infinite
number of trials.

538
00:36:24,370 --> 00:36:28,430
But that's the law of large
numbers that says the--

539
00:36:28,430 --> 00:36:36,120
Now, it's worth noting that this
law does not imply that

540
00:36:36,120 --> 00:36:42,560
if I start out with deviations
from the expected behavior,

541
00:36:42,560 --> 00:36:46,350
those deviations are likely to
be quote "evened out" by

542
00:36:46,350 --> 00:36:48,340
opposite deviations
in the future.

543
00:36:51,300 --> 00:36:54,210
So if I happen to start by
getting a whole bunch of heads

544
00:36:54,210 --> 00:36:59,960
in a row, it does not mean that
I'm more likely to get

545
00:36:59,960 --> 00:37:02,210
tails in a subsequent trial.

546
00:37:05,570 --> 00:37:06,370
All right.

547
00:37:06,370 --> 00:37:08,070
Because if I were--

548
00:37:08,070 --> 00:37:12,110
if that were true, then they
wouldn't be independent.

549
00:37:12,110 --> 00:37:14,940
Independent means memoryless.

550
00:37:14,940 --> 00:37:18,580
So if I have an independent
process, what happens in the

551
00:37:18,580 --> 00:37:22,810
future cannot be affected
by the past.

552
00:37:22,810 --> 00:37:25,910
And therefore, I don't
get this business

553
00:37:25,910 --> 00:37:29,740
of "they even out".

554
00:37:29,740 --> 00:37:32,140
Now people refuse
to believe this.

555
00:37:32,140 --> 00:37:36,140
If you go to any gambling place,
you'll discover that if

556
00:37:36,140 --> 00:37:39,080
people threw the roulette wheel,
if black comes up 20

557
00:37:39,080 --> 00:37:42,440
times in a row, they'll be
a rush to bet on red.

558
00:37:42,440 --> 00:37:46,910
Because everyone will say, red
is do, red is do, red is do.

559
00:37:46,910 --> 00:37:50,680
And every psychologist who has
ever done this experiment,

560
00:37:50,680 --> 00:37:52,920
finds that people don't
believe it.

561
00:37:52,920 --> 00:37:54,910
That it's not true.

562
00:37:54,910 --> 00:37:57,940
People just don't get
probability, and it happens so

563
00:37:57,940 --> 00:38:01,800
often it's got a name called
"the gambler's fallacy".

564
00:38:01,800 --> 00:38:06,780
And there's been great examples
of people going broke

565
00:38:06,780 --> 00:38:08,670
doing this.

566
00:38:08,670 --> 00:38:14,750
Now notice that the law of large
numbers here is about

567
00:38:14,750 --> 00:38:18,820
the fraction of times
I get an outcome.

568
00:38:18,820 --> 00:38:24,930
It does not imply for example,
that the absolute difference

569
00:38:24,930 --> 00:38:27,550
between the number of heads and
the number of tails will

570
00:38:27,550 --> 00:38:29,315
get smaller as I run
more trials.

571
00:38:32,300 --> 00:38:33,060
Right?

572
00:38:33,060 --> 00:38:36,040
It doesn't say anything
at all about that.

573
00:38:36,040 --> 00:38:40,740
It says the ratio of head to
tails will approach 1, but not

574
00:38:40,740 --> 00:38:43,360
that the difference
between them.

575
00:38:43,360 --> 00:38:46,920
All right, let's look at an
example showing that off.

576
00:38:46,920 --> 00:38:53,430
So what I've got here is this
program called "flip plot".

577
00:38:53,430 --> 00:38:54,820
This is on your hand out.

578
00:39:03,990 --> 00:39:08,710
This is just going to run this
business of flipping coins.

579
00:39:08,710 --> 00:39:10,030
I should point out just--

580
00:39:10,030 --> 00:39:12,900
I did it this way just
to show you.

581
00:39:12,900 --> 00:39:16,580
What I'm doing is each flip-- if
random.random is less than

582
00:39:16,580 --> 00:39:19,260
5, I'll call it a head,
0.5, I'll call it

583
00:39:19,260 --> 00:39:22,200
heads, otherwise a tails.

584
00:39:22,200 --> 00:39:25,160
You'll notice that it appears
that maybe I'm biasing a

585
00:39:25,160 --> 00:39:31,610
little bit, because I'm
giving 0.5 a value.

586
00:39:31,610 --> 00:39:37,250
But there are so many floating
point numbers between 0 and 1,

587
00:39:37,250 --> 00:39:41,000
that the probability of getting
exactly 0.5 is so

588
00:39:41,000 --> 00:39:43,590
small that I can ignore it.

589
00:39:43,590 --> 00:39:49,590
It isn't going to really make a
difference Random.random is

590
00:39:49,590 --> 00:39:52,900
the key issue, the key function
that's used to

591
00:39:52,900 --> 00:39:56,240
implement all the other random
functions that we

592
00:39:56,240 --> 00:39:58,980
have in that package.

593
00:39:58,980 --> 00:40:00,150
All right.

594
00:40:00,150 --> 00:40:03,700
So I'm going to do it, and
I'm going to do it

595
00:40:03,700 --> 00:40:06,030
over a range of values.

596
00:40:06,030 --> 00:40:10,030
The minimum exponent to the
maximum exponent and for

597
00:40:10,030 --> 00:40:15,710
exponent in range min x to max
x plus 1, I'm going to choose

598
00:40:15,710 --> 00:40:18,560
an x value that is 2 to that.

599
00:40:18,560 --> 00:40:21,840
So this lets me go
over a big range.

600
00:40:21,840 --> 00:40:26,090
So I'll see what happens if I
get 1 flip, and 2 flips, and 4

601
00:40:26,090 --> 00:40:31,320
and 8 and 16 and 32 et cetera.

602
00:40:31,320 --> 00:40:34,170
And then I'm going to
just do some plots.

603
00:40:34,170 --> 00:40:37,070
I'm going to plot the absolute
difference between heads and

604
00:40:37,070 --> 00:40:41,735
tails and the ratio
of heads to tails.

605
00:40:45,050 --> 00:40:49,210
Let's see what happens
when we run that.

606
00:40:49,210 --> 00:40:52,080
Actually, probably nothing
because I didn't uncomment the

607
00:40:52,080 --> 00:40:53,330
run part of it.

608
00:40:56,230 --> 00:40:57,480
Let's do that.

609
00:41:06,600 --> 00:41:12,750
So I'm going to call flip plot
with 4 and 20, running from

610
00:41:12,750 --> 00:41:18,105
four trials 2 to the
4 to 2 to the 20.

611
00:41:18,105 --> 00:41:19,355
Let's see what we get.

612
00:41:24,430 --> 00:41:29,120
Now, you may get different
things when you run at

613
00:41:29,120 --> 00:41:30,190
different times.

614
00:41:30,190 --> 00:41:31,440
In fact, you will.

615
00:41:34,270 --> 00:41:39,440
So here we see something
kind of uninteresting.

616
00:41:42,580 --> 00:41:47,290
Let's cheat and see what we got
the first time I ran it,

617
00:41:47,290 --> 00:41:52,220
which is on your hand out, and
I have a PowerPoint with it.

618
00:41:52,220 --> 00:41:53,360
I was--

619
00:41:53,360 --> 00:41:54,960
I knew this might happen.

620
00:41:54,960 --> 00:41:57,910
Doesn't usually, but sometimes
when you run it you get

621
00:41:57,910 --> 00:41:59,160
surprising results.

622
00:42:02,310 --> 00:42:04,730
So here's what happened
when I first ran it.

623
00:42:08,600 --> 00:42:10,645
Here was the difference between
heads and tails.

624
00:42:13,360 --> 00:42:16,620
And it seems that, OK, the
difference was low, it went

625
00:42:16,620 --> 00:42:20,280
up, it went down, it went
up, it went down.

626
00:42:20,280 --> 00:42:21,810
It seemed to go down
dramatically.

627
00:42:26,310 --> 00:42:30,590
If you remember what we just saw
when I ran it, we also saw

628
00:42:30,590 --> 00:42:33,140
something where it went up a
little bit then it went down

629
00:42:33,140 --> 00:42:35,790
and then shot up dramatically at
the end, which was why that

630
00:42:35,790 --> 00:42:37,970
scale is so funny.

631
00:42:37,970 --> 00:42:41,890
And if we look at the ratio,
what we see is it seems to

632
00:42:41,890 --> 00:42:47,490
start above 1, drop below
1, and then seems to

633
00:42:47,490 --> 00:42:50,920
converge towards 1.

634
00:42:50,920 --> 00:42:53,520
Now, I show this because I
want to make a couple of

635
00:42:53,520 --> 00:42:54,770
points of plotting.

636
00:42:57,460 --> 00:43:00,990
Let's look at this out here.

637
00:43:00,990 --> 00:43:04,590
Looks like we have a pretty
dramatic trend

638
00:43:04,590 --> 00:43:07,920
of this linear drop.

639
00:43:07,920 --> 00:43:10,140
Do we?

640
00:43:10,140 --> 00:43:11,645
Do we actually have
a trend here?

641
00:43:16,940 --> 00:43:19,950
Well let's think about it.

642
00:43:19,950 --> 00:43:25,690
The default behavior of the plot
command in PyLab is to

643
00:43:25,690 --> 00:43:27,100
connect points by lines.

644
00:43:30,160 --> 00:43:32,580
How many points do I actually
have out here?

645
00:43:41,770 --> 00:43:43,230
Well, you saw the code.

646
00:43:43,230 --> 00:43:45,650
You have the code
in the hand out.

647
00:43:45,650 --> 00:43:47,800
How many points do you think
there are out here?

648
00:43:50,674 --> 00:43:53,548
A 1,000?

649
00:43:53,548 --> 00:43:56,430
A 100?

650
00:43:56,430 --> 00:43:58,110
3?

651
00:43:58,110 --> 00:44:00,550
2?

652
00:44:00,550 --> 00:44:00,830
2 to 3.

653
00:44:00,830 --> 00:44:01,630
Right?

654
00:44:01,630 --> 00:44:04,360
Depending on what I mean
by "out here".

655
00:44:04,360 --> 00:44:08,590
So what we see here is something
that happens a lot.

656
00:44:08,590 --> 00:44:12,120
People plot a small number of
points, connect them by a

657
00:44:12,120 --> 00:44:17,450
line, and mislead the audience
into thinking there's a trend

658
00:44:17,450 --> 00:44:19,435
when, in fact, maybe all
you have is an outlier.

659
00:44:22,940 --> 00:44:31,060
So it's problematical here
to do it that way.

660
00:44:31,060 --> 00:44:33,480
So let's see what happens
if we change the code.

661
00:44:58,170 --> 00:45:09,550
And what I'm going to do is
change it in two ways.

662
00:45:12,830 --> 00:45:15,310
Well, maybe I'll change
it in one way first.

663
00:45:26,290 --> 00:45:28,950
Uncomment.

664
00:45:28,950 --> 00:45:30,200
Uncomment.

665
00:45:44,400 --> 00:45:50,550
So what I'm doing here is I am
plotting in a different way.

666
00:45:50,550 --> 00:45:56,390
This quote "BO" says, don't
connect it by lines but just

667
00:45:56,390 --> 00:46:01,190
put a dot as an "O" and
B says, make it blue.

668
00:46:01,190 --> 00:46:04,030
I used blue because it's
my favorite color.

669
00:46:04,030 --> 00:46:06,600
So now if we look at these
things, we'll see something

670
00:46:06,600 --> 00:46:07,850
pretty different.

671
00:46:17,280 --> 00:46:20,816
So that's the difference between
heads and tails,

672
00:46:20,816 --> 00:46:23,420
that's the ratio.

673
00:46:23,420 --> 00:46:27,070
But now, if we look at the
difference between heads and

674
00:46:27,070 --> 00:46:32,610
tails here, what we see
is it's pretty sparse.

675
00:46:32,610 --> 00:46:36,850
So yeah, maybe there's a trend,
but maybe not, right?

676
00:46:36,850 --> 00:46:41,730
Because way out here I'm only
connecting two points, giving

677
00:46:41,730 --> 00:46:47,065
an illusion that there is a
trend but, in fact, no reason

678
00:46:47,065 --> 00:46:48,880
to believe it.

679
00:46:48,880 --> 00:46:51,700
So I always think if you're
plotting a small number of

680
00:46:51,700 --> 00:46:56,410
points, you're much better off
just plotting the points, than

681
00:46:56,410 --> 00:47:01,820
you are trying to
connect them.

682
00:47:01,820 --> 00:47:06,400
Now if we look at this one,
again, maybe we'd feel kind of

683
00:47:06,400 --> 00:47:08,400
comfortable if there is a trend
here, that there are

684
00:47:08,400 --> 00:47:10,530
several points on this line.

685
00:47:10,530 --> 00:47:16,540
We can't see much of what's
going on over here, which gets

686
00:47:16,540 --> 00:47:27,780
me to the next thing I want
to do is I'm going to use

687
00:47:27,780 --> 00:47:29,530
logarithmic axes here.

688
00:47:32,360 --> 00:47:37,900
So PyLab.semilogx says make
the x-axis logarithmic.

689
00:47:37,900 --> 00:47:41,730
PyLab.semilogy the y-axis.

690
00:47:41,730 --> 00:47:45,710
And so in the case of the
absolute difference, I'm going

691
00:47:45,710 --> 00:47:48,460
to make both logarithmic.

692
00:47:48,460 --> 00:47:50,220
Why am I doing that?

693
00:47:50,220 --> 00:47:56,270
Because both have a large
range, and by making it

694
00:47:56,270 --> 00:47:59,450
logarithmic, I can see what's
happening at the left side in

695
00:47:59,450 --> 00:48:03,570
this case where things
are changing.

696
00:48:03,570 --> 00:48:07,160
When I look at the ratios, the
y-axis does not have a very

697
00:48:07,160 --> 00:48:10,645
large range, and so there's no
need to make it logarithmic.

698
00:48:13,540 --> 00:48:14,790
We'll run it.

699
00:48:24,840 --> 00:48:27,750
So here, we can see
the difference

700
00:48:27,750 --> 00:48:29,880
between heads and tails.

701
00:48:29,880 --> 00:48:33,610
And now we can see what's going
on at the left as we can

702
00:48:33,610 --> 00:48:34,860
in Figure (4).

703
00:48:38,690 --> 00:48:42,380
And we can see things
much more clearly.

704
00:48:42,380 --> 00:48:45,770
So log scales can be
enormously useful.

705
00:48:48,810 --> 00:48:53,190
And in fact, I use them a lot,
everyone uses them a lot but,

706
00:48:53,190 --> 00:48:57,920
again, it's very important to
observe the fact that it's

707
00:48:57,920 --> 00:49:00,180
logarithmic and not
get fooled.

708
00:49:05,570 --> 00:49:06,820
All right.

709
00:49:09,120 --> 00:49:14,290
So I talked about linear
scaling, logarithmic scaling

710
00:49:14,290 --> 00:49:19,340
and we now have charts where I
can, perhaps, actually reach

711
00:49:19,340 --> 00:49:23,415
some conclusion about
what's going on.

712
00:49:23,415 --> 00:49:28,500
The next question is, how
certain can I be?

713
00:49:28,500 --> 00:49:33,090
Can I really be certain that,
indeed, this should be

714
00:49:33,090 --> 00:49:34,340
converging to 1?

715
00:49:37,140 --> 00:49:40,330
Here, if I sort of look at it,
it does look like there's kind

716
00:49:40,330 --> 00:49:45,380
of a linear trend of the
absolute difference growing as

717
00:49:45,380 --> 00:49:47,400
the number of trials grows.

718
00:49:47,400 --> 00:49:51,540
How certain can I be of that?

719
00:49:51,540 --> 00:49:56,700
You can never get absolute
certainty from sampling,

720
00:49:56,700 --> 00:49:59,140
because you could never be
sure if you haven't been

721
00:49:59,140 --> 00:50:02,170
vastly lucky or unlucky.

722
00:50:02,170 --> 00:50:04,630
That's not to say you
can't get the

723
00:50:04,630 --> 00:50:06,260
absolute correct answer.

724
00:50:06,260 --> 00:50:12,160
Maybe I could get 0.5, which
is the correct answer.

725
00:50:12,160 --> 00:50:17,180
But I can't know that that's
the correct answer.

726
00:50:17,180 --> 00:50:21,290
So now the question I want to
pursue, and it's what we'll

727
00:50:21,290 --> 00:50:27,520
cover on Thursday, is what
techniques can I use to make a

728
00:50:27,520 --> 00:50:31,650
statement of the form, I'm
certain within the following

729
00:50:31,650 --> 00:50:34,105
range that I have the
right answer.

730
00:50:34,105 --> 00:50:38,450
That I know the right answer
is highly likely to be this

731
00:50:38,450 --> 00:50:41,450
close to the answer my
simulation is giving me.

732
00:50:41,450 --> 00:50:44,410
And we'll look at how we can
make those statements and

733
00:50:44,410 --> 00:50:46,140
actually believe them.

734
00:50:46,140 --> 00:50:47,390
OK, see you on Thursday.