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PROFESSOR: So, if you remember,
just before the break, as long

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00:00:26 --> 00:00:31
ago as it was, we had looked at
the problem of fitting

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curves to data.


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And the example we had seen, is
that it's often possible, in

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fact, usually possible, to find
a good fit to old values.

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00:00:49 --> 00:00:51
What we looked at was, we
looked at a small number of

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00:00:51 --> 00:00:54
points, we took a high degree
polynomial, sure enough,

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we got a great fit.


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00:00:57 --> 00:01:02
The difficulty was, a great
fit to old values does

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not necessarily imply a
good fit to new values.

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00:01:24 --> 00:01:31
And in general, that's
somewhat worrisome.

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00:01:31 --> 00:01:34
So now I want to spend a little
bit of time I'm looking at some

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tools, that we can use to
better understand the notion

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of, when we have a bunch of
points, what do they look like?

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How does the variation work?


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00:01:47 --> 00:01:51
This gets back to a concept
that we've used a number of

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times, which is a notion
of a distribution.

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00:01:59 --> 00:02:04
Remember, the whole logic
behind our idea of using

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00:02:04 --> 00:02:11
simulation, or polling, or any
kind of statistical technique,

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is the assumption that the
values we would draw were

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representative of the values
of the larger population.

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00:02:21 --> 00:02:25
We're sampling some subset of
the population, and we're

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assuming that that sample is
representative of the

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greater population.


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We talked about several
different issues

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related to that.


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00:02:36 --> 00:02:40
I now want to look at that a
little bit more formally.

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00:02:40 --> 00:02:47
And we'll start with the very
old problem of rolling dice.

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00:02:47 --> 00:02:50
I presume you've all seen
what a pair of dice

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00:02:50 --> 00:02:51
look like, right?


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00:02:51 --> 00:02:54
They've got the numbers 1
through 6 on them, you roll

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them and something comes up.


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00:02:57 --> 00:03:01
If you haven't seen it, if you
look at the very back, at the

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back page of the handout today,
you'll see a picture

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of a very old die.


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00:03:06 --> 00:03:11
Some time from the fourth
to the second century BC.

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00:03:11 --> 00:03:15
Looks remarkably like a modern
dice, except it's not made

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out of plastic, it's
made out of bones.

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00:03:18 --> 00:03:20
And in fact, if you were
interested in the history of

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gambling, or if you happen to
play with dice, people

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do call them bones.


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00:03:25 --> 00:03:28
And that just dates back to
the fact that the original

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ones were made that way.


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00:03:30 --> 00:03:35
And in fact, what we'll see is,
that in the history of

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probability and statistics, an
awful lot of the math that we

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00:03:39 --> 00:03:42
take for granted today, came
from people's attempts to

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understand various
games of chance.

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00:03:46 --> 00:03:50
So, let's look at it.


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00:03:50 --> 00:03:53
So we'll look at this program.


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You should have this in
the front of the handout.

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So I'm going to start
with a fair dice.

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That is to say, when you roll
it, it's equally probable that

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you get 1, 2, 3, 4, 5, or 6.


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And I'm going to throw a pair.


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You can see it's very simple.


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I'll take d 1, first die is
random dot choice from vals

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1. d 2 will be random
dot choice from vals 2.

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So I'm going to pass it in two
sets of possible values, and

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randomly choose one or the
other, and then return them.

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And the way I'll conduct a
trial is, I'll take some

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number of throws, and two
different kinds of dice.

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Throws will be the empty
set, actually, yeah.

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And then I'll just do it.


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For i in range number of
throws, d 1, d 2 is equal to

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throw a pair, and then I'll
append it, and then

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I'll return it.


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Very simple, right?


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Could hardly imagine a
simpler little program.

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And then, we'll analyze it.


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And we're going to analyze it.


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Well, first let's analyze it
one way, and then we'll look at

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00:05:25 --> 00:05:41
something slightly different.


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I'm going to conduct
some number of trials

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with two fair die.


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Then I'm going to make a
histogram, because I happen

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to know that there are
only 11 possible values,

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I'll make 11 bins.


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You may not have seen
this locution here,

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Pylab dot x ticks.


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That's telling it where to put
the markers on the x-axis,

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and what they should be.


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In this case 2 through 12,
and then I'll label it.

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So let's run this program.


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And here we see the
distribution of values.

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So we see that I get more
7s than anything else,

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00:06:38 --> 00:06:42
and fewer 2s and 12s.


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00:06:42 --> 00:06:47
Snake eyes and boxcars
to you gamblers.

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00:06:47 --> 00:06:51
And it's a beautiful
distribution, in some sense.

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00:06:51 --> 00:06:53
I ran it enough trials.


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This kind of distribution
is called normal.

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00:07:01 --> 00:07:07
Also sometimes called Gaussian,
after the mathematician Gauss.

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00:07:07 --> 00:07:11
Sometimes called the
bell curve, because in

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someone's imagination
it looks like a bell.

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We see these things
all the time.

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00:07:22 --> 00:07:27
They're called normal, or
sometimes even natural, because

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it's probably the most commonly
observed probability

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00:07:30 --> 00:07:35
distribution in nature.


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00:07:35 --> 00:07:38
First documented, although
I'm sure not first

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00:07:38 --> 00:07:44
seen, by deMoivre and
Laplace in the 1700s.

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00:07:44 --> 00:07:48
And then in the 1800s,
Gauss used it to analyze

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00:07:48 --> 00:07:51
astronomical data.


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And it got to be called,
in that case, the

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00:07:53 --> 00:07:57
Gaussian distribution.


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00:07:57 --> 00:07:58
So where do we see it occur?


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We see it occurring
all over the place.

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00:08:00 --> 00:08:04
We certainly see
it rolling dice.

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00:08:04 --> 00:08:09
We see it occur in things like
the distribution of heights.

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00:08:09 --> 00:08:12
If we were to take the height
of all the students at MIT and

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00:08:12 --> 00:08:17
plot the distribution, I would
be astonished if it didn't

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00:08:17 --> 00:08:18
look more or less like that.


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00:08:18 --> 00:08:22
It would be a normal
distribution.

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00:08:22 --> 00:08:24
A lot of things in
the same height.

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00:08:24 --> 00:08:26
Now presumably, we'd have to
round off to the nearest

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millimeter or something.


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And a few really tall people,
and a few really short people.

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It's just astonishing
in nature how often we

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00:08:38 --> 00:08:40
look at these things.


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The graph looks exactly like
that, or similar to that.

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The shape is roughly that.


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00:08:47 --> 00:08:53
The normal distribution can
be described, interestingly

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enough, with just two numbers.


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The mean and the
standard deviation.

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So if I give you those
two numbers, you can

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00:09:16 --> 00:09:18
draw that curve.


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Now you might not be able to
label, you couldn't label the

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00:09:21 --> 00:09:25
axes, because how would you
know how many trials

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00:09:25 --> 00:09:26
I did, right?


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00:09:26 --> 00:09:29
Whether I did 100, or 1,000
or a million, but the shape

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00:09:29 --> 00:09:34
would always be the same.


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00:09:34 --> 00:09:39
And if I were to, instead of
doing, 100,000 throws of the

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00:09:39 --> 00:09:44
dice, as I did here, I did a
million, the label on the

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00:09:44 --> 00:09:47
y-axis would change,
but the shape would be

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absolutely identical.


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00:09:51 --> 00:09:54
This is what's called a
stable distribution.

143
00:09:54 --> 00:10:14
As you change the scale,
the shape doesn't change.

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00:10:14 --> 00:10:19
So the mean tells us where it's
centered, and the standard

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00:10:19 --> 00:10:23
deviation, basically, is a
measure of statistical

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00:10:23 --> 00:10:42
dispersion.


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It tells us how widely spread
the points of the data set are.

148
00:10:48 --> 00:10:52
If many points are going to be
very close to the mean, then

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the standard deviation
is what, big or small?

150
00:10:57 --> 00:10:59
Pardon?


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Small.


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00:11:04 --> 00:11:09
If they're spread out, and it's
kind of a flat bell, then the

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00:11:09 --> 00:11:13
standard deviation
will be large.

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00:11:13 --> 00:11:16
And I'm sure you've all
seen standard deviations.

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We give exams, and we say
here's the mean, here's

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00:11:19 --> 00:11:22
the standard deviation.


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00:11:22 --> 00:11:25
And the notion is, that's
trying to tell you what the

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00:11:25 --> 00:11:29
average score was, and
how spread out they are.

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00:11:29 --> 00:11:35
Now as it happens, rarely do
we have exams that actually

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00:11:35 --> 00:11:37
fall on a bell curve.


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00:11:37 --> 00:11:38
Like this.


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00:11:38 --> 00:11:44
So in a way, don't be deceived
by thinking that we're really

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00:11:44 --> 00:11:48
giving you a good measure of
the dispersion, in the

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00:11:48 --> 00:11:52
sense, that we would get
with the bell curve.

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00:11:52 --> 00:11:58
So the standard deviation does
have a formal value, usually

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00:11:58 --> 00:12:11
written sigma, and it's the
estimates of x squared minus

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00:12:11 --> 00:12:19
the estimates of x, and then I
take all of this, the estimates

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00:12:19 --> 00:12:26
of x, right, squared.


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00:12:26 --> 00:12:29
So I don't worry much about
this, but what I'm basically

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00:12:29 --> 00:12:35
doing is, x is all of
the values I have.

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00:12:35 --> 00:12:42
And I can square each of the
values, and then I subtract

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00:12:42 --> 00:12:50
from that, the sum of the
values, squaring that.

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00:12:50 --> 00:12:56
What's more important than this
formula, for most uses, is what

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00:12:56 --> 00:13:03
people think of as the --
why didn't I write it down

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00:13:03 --> 00:13:07
-- this is interesting.


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00:13:07 --> 00:13:10
There is a some number -- I see
why, I did write it down, it

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just got printed on two-sided.


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00:13:12 --> 00:13:26
Is the Empirical Rule.


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00:13:26 --> 00:13:35
And this applies for
normal distributions.

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00:13:35 --> 00:13:38
So anyone know how much of the
data that you should expect to

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00:13:38 --> 00:13:44
fall within one standard
deviation of the mean?

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00:13:44 --> 00:13:49
68.


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00:13:49 --> 00:13:59
So 68% within one, 95% of
the data falls within two,

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00:13:59 --> 00:14:06
and almost all of the
data within three.

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00:14:06 --> 00:14:11
These values are
approximations, by the way.

186
00:14:11 --> 00:14:20
So, this is, really 95%
falls within 1.96 standard

187
00:14:20 --> 00:14:23
deviations, it's not two.


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00:14:23 --> 00:14:28
But this gives you a sense
of how spread out it is.

189
00:14:28 --> 00:14:33
And again, this applies only
for a normal distribution.

190
00:14:33 --> 00:14:37
If you compute the standard
deviation this way, and apply

191
00:14:37 --> 00:14:40
it to something other than a
normal distribution, there's

192
00:14:40 --> 00:14:52
no reason to expect that
Empirical Rule will hold.

193
00:14:52 --> 00:14:55
OK people with me on this?


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00:14:55 --> 00:14:58
It's amazing to me how many
people in society talk about

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standard deviations without
actually knowing what they are.

196
00:15:02 --> 00:15:05
And there's another way to
look at the same data,

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00:15:05 --> 00:15:07
or almost the same data.


198
00:15:07 --> 00:15:22
Since it's a random experiment,
it won't be exactly the same.

199
00:15:22 --> 00:15:28
So as before, we had the
distribution, and fortunately

200
00:15:28 --> 00:15:30
it looks pretty much
like the last one.

201
00:15:30 --> 00:15:32
We would've expected that.


202
00:15:32 --> 00:15:35
And I've now done something,
another way of looking at the

203
00:15:35 --> 00:15:41
same information, really, is,
I printed, I plotted,

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00:15:41 --> 00:15:46
the probabilities of
different values.

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00:15:46 --> 00:15:50
So we can see here that the
probability of getting

206
00:15:50 --> 00:15:59
a 7 is about 0.17 or
something like that.

207
00:15:59 --> 00:16:03
Now, since I threw 100,000 die,
it's not surprising that the

208
00:16:03 --> 00:16:11
probability of 0.17 looks about
the same as 17,000 over here.

209
00:16:11 --> 00:16:15
But it's just a different
way of looking at things.

210
00:16:15 --> 00:16:21
Right, had I thrown some
different number, it might

211
00:16:21 --> 00:16:23
have been harder to visualize
what the probability

212
00:16:23 --> 00:16:26
distribution looked like.


213
00:16:26 --> 00:16:28
But we often do
talk about that.

214
00:16:28 --> 00:16:35
How probable is a
certain value?

215
00:16:35 --> 00:16:39
People who design games of
chance, by the way, something

216
00:16:39 --> 00:16:39
I've been meaning to say.


217
00:16:39 --> 00:16:43
You'll notice down here there's
just, when we want to save

218
00:16:43 --> 00:16:46
these things, there's this
little icon that's a floppy

219
00:16:46 --> 00:16:48
disk, to indicate store.


220
00:16:48 --> 00:16:52
And I thought maybe many of
you'd never seen a floppy disk,

221
00:16:52 --> 00:16:53
so I decided to bring one in.


222
00:16:53 --> 00:16:55
You've seen the icons.


223
00:16:55 --> 00:16:58
And probably by the time most
of you got, any you ever

224
00:16:58 --> 00:17:01
had a machine with a
quote floppy drive?

225
00:17:01 --> 00:17:04
Did it actually flop the disk?


226
00:17:04 --> 00:17:07
No, they were pretty rigid,
but the old floppy disks

227
00:17:07 --> 00:17:09
were really floppy.


228
00:17:09 --> 00:17:12
Hence they got the name.


229
00:17:12 --> 00:17:16
And, you know it's kind of
like a giant size version.

230
00:17:16 --> 00:17:19
And it's amazing how people
will probably continue to

231
00:17:19 --> 00:17:22
talk about floppy disks as
long as they talk about

232
00:17:22 --> 00:17:24
dialing a telephone.


233
00:17:24 --> 00:17:27
And probably none of you've
ever actually dialed a phone,

234
00:17:27 --> 00:17:29
for that matter, just
pushed buttons .

235
00:17:29 --> 00:17:33
But they used to have dials
that you would twirl.

236
00:17:33 --> 00:17:34
Anyway, I just thought
everyone should at least

237
00:17:34 --> 00:17:37
see a floppy disk once.


238
00:17:37 --> 00:17:40
This is, by the way, a very
good way to get data security.

239
00:17:40 --> 00:17:42
There's probably no way in the
world to read the information

240
00:17:42 --> 00:17:48
on this disk anymore.


241
00:17:48 --> 00:17:54
All right, as I said people
who design games of

242
00:17:54 --> 00:17:58
chance understand these
probabilities very well.

243
00:17:58 --> 00:18:02
So I'm gonna now look at, show
how we can understand these

244
00:18:02 --> 00:18:06
things in some other ways
of popular example.

245
00:18:06 --> 00:18:09
A game of dice.


246
00:18:09 --> 00:18:13
Has anyone here ever played
the game called craps?

247
00:18:13 --> 00:18:16
Did you win or lose money?


248
00:18:16 --> 00:18:18
You won.


249
00:18:18 --> 00:18:21
All right, you beat the odds.


250
00:18:21 --> 00:18:23
Well, it's a very popular
game, and I'm going

251
00:18:23 --> 00:18:25
to explain it to you.


252
00:18:25 --> 00:18:28
As you will see, this is not
an endorsement of gambling,

253
00:18:28 --> 00:18:30
because one of the things you
will notice is, you are likely

254
00:18:30 --> 00:18:32
to lose money if you do this.


255
00:18:32 --> 00:18:35
So I tend not, I don't do it.


256
00:18:35 --> 00:18:45
All right, so how does
a game of craps work?

257
00:18:45 --> 00:18:50
You start by rolling two dice.


258
00:18:50 --> 00:18:55
If you get a 7 or an 11, the
roller, we'll call that

259
00:18:55 --> 00:19:01
the shooter, you win.


260
00:19:01 --> 00:19:08
If you get a 2, 3,
or a 12, you lose.

261
00:19:08 --> 00:19:11
I'm assuming here, you're
betting what's called

262
00:19:11 --> 00:19:12
the pass line.


263
00:19:12 --> 00:19:14
There are different ways
to bet, this is the most

264
00:19:14 --> 00:19:17
common way to bet, we'll
just deal with that.

265
00:19:17 --> 00:19:27
If it's not any of these, what
you get is, otherwise, the

266
00:19:27 --> 00:19:37
number becomes what's
called the point.

267
00:19:37 --> 00:19:41
Once you've got the point, you
keep rolling the dice until

268
00:19:41 --> 00:19:44
1 of o things happens.


269
00:19:44 --> 00:19:56
You get a 7, in which case you
lose, or you get the point,

270
00:19:56 --> 00:20:04
in which case you win.


271
00:20:04 --> 00:20:07
So it's a pretty simple game.


272
00:20:07 --> 00:20:12
Very popular game.


273
00:20:12 --> 00:20:14
So I've implemented.


274
00:20:14 --> 00:20:18
So one of the interesting
things about this is, if you

275
00:20:18 --> 00:20:22
try and actually figure out
what the probabilities are

276
00:20:22 --> 00:20:25
using pencil and paper,
you can, but it gets a

277
00:20:25 --> 00:20:33
little bit involved.


278
00:20:33 --> 00:20:35
Gets involved because you have
to, all right, what are the

279
00:20:35 --> 00:20:38
odds of winning or losing
on the first throw?

280
00:20:38 --> 00:20:41
Well, you can compute those
pretty easily, and you can see

281
00:20:41 --> 00:20:43
that you'd actually win more
than you lose, on

282
00:20:43 --> 00:20:44
the first throw.


283
00:20:44 --> 00:20:48
But if you look at the
distribution of 7s and 11s and

284
00:20:48 --> 00:20:51
2s, 3s, and 12s, you add them
up, you'll see, well, this is

285
00:20:51 --> 00:20:53
more likely than this.


286
00:20:53 --> 00:20:55
But then you say, suppose
I don't get those.

287
00:20:55 --> 00:20:58
What's the likelihood of
getting each other possible

288
00:20:58 --> 00:21:02
point value, and then given
that point value, what's the

289
00:21:02 --> 00:21:04
likelihood of getting
that before a 7?

290
00:21:04 --> 00:21:07
And you can do it, but
it gets very tedious.

291
00:21:07 --> 00:21:13
So those of us who are inclined
to think computationally, and I

292
00:21:13 --> 00:21:17
hope by now that's all of you
as well as me, say well,

293
00:21:17 --> 00:21:20
instead of doing the
probabilities by hand, I'm just

294
00:21:20 --> 00:21:22
going to write a
little program.

295
00:21:22 --> 00:21:27
And it's a program that took
me maybe 10 minutes to write.

296
00:21:27 --> 00:21:34
You can see it's quite
small, I did it yesterday.

297
00:21:34 --> 00:21:39
So the first function here is
craps, it returns true if the

298
00:21:39 --> 00:21:43
shooter wins by betting
the pass line.

299
00:21:43 --> 00:21:46
And it's just does what I said.


300
00:21:46 --> 00:21:49
Rolls them, if the total
is 1 or 11, it returns

301
00:21:49 --> 00:21:51
true, you win.


302
00:21:51 --> 00:21:54
If it's 2, 3, or 12, it
returns false, you lose.

303
00:21:54 --> 00:21:58
Otherwise the point
becomes the total.

304
00:21:58 --> 00:22:01
And then while true,
I'll just keep rolling.

305
00:22:01 --> 00:22:07
Until either, if the total gets
the point, I return true.

306
00:22:07 --> 00:22:09
Or if the total's equal
7, I return false.

307
00:22:09 --> 00:22:12
And that's it.


308
00:22:12 --> 00:22:15
So essentially I just took
these rules, typed them

309
00:22:15 --> 00:22:20
down, and I had my game.


310
00:22:20 --> 00:22:24
And then I'll simulate it
will some number of bets.

311
00:22:24 --> 00:22:28
Keeping track of the numbers
of wins and losses.

312
00:22:28 --> 00:22:31
Just by incrementing 1 or the
other, depending upon whether

313
00:22:31 --> 00:22:34
I return true or false.


314
00:22:34 --> 00:22:36
I'm going to, just to show
what we do, print the

315
00:22:36 --> 00:22:39
number of wins and losses.


316
00:22:39 --> 00:22:44
And then compute, how
does the house do?

317
00:22:44 --> 00:22:47
Not the gambler, but the
person who's running

318
00:22:47 --> 00:22:49
the game, the casino.


319
00:22:49 --> 00:22:53
Or in other circumstances,
other places.

320
00:22:53 --> 00:22:56
And then we'll see
how that goes.

321
00:22:56 --> 00:23:00
And I'll try it with
100,000 games.

322
00:23:00 --> 00:23:03
Now, this is more than 100,000
rolls of the dice, right?

323
00:23:03 --> 00:23:08
Because I don't get a 7
or 11, I keep rolling.

324
00:23:08 --> 00:23:15
So before I do it, I'll as
the easy question first.

325
00:23:15 --> 00:23:21
Who thinks the casino wins
more often than the player?

326
00:23:21 --> 00:23:26
Who thinks the player wins
more often than the casino?

327
00:23:26 --> 00:23:29
Well, very logical, casinos
are not in business

328
00:23:29 --> 00:23:32
of giving away money.


329
00:23:32 --> 00:23:34
So now the more
interesting question.

330
00:23:34 --> 00:23:38
How steep do you think the odds
are in the house's favor?

331
00:23:38 --> 00:23:43
Anyone want to guess?


332
00:23:43 --> 00:23:45
Actually pretty thin.


333
00:23:45 --> 00:23:57
Let's run it and see.


334
00:23:57 --> 00:24:00
So what we see is, the house
wins 50, in this case

335
00:24:00 --> 00:24:04
50.424% of the time.


336
00:24:04 --> 00:24:06
Not a lot.


337
00:24:06 --> 00:24:08
On the other hand, if
people bet 100,000,

338
00:24:08 --> 00:24:14
the house wins 848.


339
00:24:14 --> 00:24:23
Now, 100,000 is actually
a small number.

340
00:24:23 --> 00:24:25
Let's get rid of these, should
have gotten rid of these

341
00:24:25 --> 00:24:53
figures, you don't need
to see them every time.

342
00:24:53 --> 00:24:58
We'll keep one figure,
just for fun.

343
00:24:58 --> 00:25:04
Let's try it again.


344
00:25:04 --> 00:25:06
Probably get a little
different answer.

345
00:25:06 --> 00:25:11
Considerable different, but
still, less than 51% of

346
00:25:11 --> 00:25:13
the time in this trial.


347
00:25:13 --> 00:25:16
But you can see that the house
is slowly but surely going to

348
00:25:16 --> 00:25:20
get rich playing this game.


349
00:25:20 --> 00:25:24
Now let's ask the other
interesting question.

350
00:25:24 --> 00:25:29
Just for fun, suppose
we want to cheat.

351
00:25:29 --> 00:25:32
Now, I realize none of
you would never do that.

352
00:25:32 --> 00:25:36
But let's consider using
a pair of loaded dice.

353
00:25:36 --> 00:25:40
So there's a long history, well
you can imagine when you looked

354
00:25:40 --> 00:25:44
at that old bone I showed you,
that it wasn't exactly fair.

355
00:25:44 --> 00:25:47
That some sides were a little
heavier than others, and in

356
00:25:47 --> 00:25:52
fact you didn't get, say, a
5 exactly 1/6 of the time.

357
00:25:52 --> 00:25:55
And therefore, if you were
using your own dice, instead of

358
00:25:55 --> 00:25:57
somebody else's, and you knew
what was most likely,

359
00:25:57 --> 00:26:00
you might do better.


360
00:26:00 --> 00:26:02
Well, the modern version of
that is, people do cheat by

361
00:26:02 --> 00:26:07
putting little weights in dice,
to just make tiny changes in

362
00:26:07 --> 00:26:11
the probability of one number
or another coming up.

363
00:26:11 --> 00:26:14
So let's do that.


364
00:26:14 --> 00:26:16
And let's first ask the
question, well, what would

365
00:26:16 --> 00:26:20
be a nice way to do that?


366
00:26:20 --> 00:26:24
It's very easy here.


367
00:26:24 --> 00:26:37
If we look at it, all I've
done is, I've changed the

368
00:26:37 --> 00:26:40
distribution of values, so
instead of here being 1, 2,

369
00:26:40 --> 00:26:46
3, 4, 5, and 6, it's 1,
2, 3, 4, 5, 5, and 6.

370
00:26:46 --> 00:26:51
I snuck in an extra 5 on
one of the two dice.

371
00:26:51 --> 00:26:57
So this has changed the odds
of rolling a 5 from 1 in

372
00:26:57 --> 00:27:04
6 to roughly 3 in 12.


373
00:27:04 --> 00:27:09
Now 1/6, which is 2/12,
vs. 3/12, it's not

374
00:27:09 --> 00:27:10
a big difference.


375
00:27:10 --> 00:27:13
And you can imagine, if you
were sitting there watching it,

376
00:27:13 --> 00:27:17
you wouldn't notice that 5 was
coming up a little bit more

377
00:27:17 --> 00:27:19
often than you expected.


378
00:27:19 --> 00:27:21
Normally.


379
00:27:21 --> 00:27:23
Close enough that you
wouldn't notice it.

380
00:27:23 --> 00:27:28
But let's see if, what
difference it makes?

381
00:27:28 --> 00:27:30
What difference do you
think it will make here?

382
00:27:30 --> 00:27:36
First of all, is it going to be
better or worse for the player?

383
00:27:36 --> 00:27:39
Who thinks better?


384
00:27:39 --> 00:27:42
Who thinks worse?


385
00:27:42 --> 00:27:45
Who thinks they haven't a clue?


386
00:27:45 --> 00:27:48
All right, we have
an honest man.

387
00:27:48 --> 00:27:52
Where is Diogenes
when we we need him?

388
00:27:52 --> 00:27:55
The reward for honesty.


389
00:27:55 --> 00:27:58
I could reward you and wake
him up at the same time.

390
00:27:58 --> 00:28:01
It's good.


391
00:28:01 --> 00:28:16
All right, well, let's
see what happens.

392
00:28:16 --> 00:28:20
All right, so suddenly,
the odds have swung in

393
00:28:20 --> 00:28:23
favor of the player.


394
00:28:23 --> 00:28:28
This tiny little change has now
made it likely that the player

395
00:28:28 --> 00:28:33
win money, instead
of the house.

396
00:28:33 --> 00:28:34
So what's the point?


397
00:28:34 --> 00:28:38
The point is not, you should go
out and try and cheat casinos,

398
00:28:38 --> 00:28:42
because you'll probably find an
unpleasant consequence of that.

399
00:28:42 --> 00:28:47
The point is that, once I've
written this simulation, I

400
00:28:47 --> 00:28:52
can play thought experiments
in a very easy way.

401
00:28:52 --> 00:28:53
So-called what if games.


402
00:28:53 --> 00:28:54
What if we did this?


403
00:28:54 --> 00:28:56
What if we did that?


404
00:28:56 --> 00:28:59
And it's trivial to do
those kinds of things.

405
00:28:59 --> 00:29:02
And that's one of the reasons
we typically do try and

406
00:29:02 --> 00:29:04
write these simulations.


407
00:29:04 --> 00:29:09
So that we can
experiment with things.

408
00:29:09 --> 00:29:11
Are there any other experiments
people would like to

409
00:29:11 --> 00:29:13
perform while we're here?


410
00:29:13 --> 00:29:18
Any other sets of die
you might like to try?

411
00:29:18 --> 00:29:22
All right, someone give me a
suggestion of something that

412
00:29:22 --> 00:29:24
might work in the
house's favor.

413
00:29:24 --> 00:29:27
Suppose a casino
wanted to cheat.

414
00:29:27 --> 00:29:28
What do you think
would help them out?

415
00:29:28 --> 00:29:28
Yeah?


416
00:29:28 --> 00:29:32
STUDENT: Increase prevalence
of 1, instead of 5?

417
00:29:32 --> 00:29:34
PROFESSOR: All right, so let's
see if we increase the

418
00:29:34 --> 00:29:52
probability of 1, what it does?


419
00:29:52 --> 00:29:58
Yep, clearly helped the
house out, didn't it?

420
00:29:58 --> 00:30:04
So that would be a good
thing for the house.

421
00:30:04 --> 00:30:08
Again, you know, three key
strokes and we get to try it.

422
00:30:08 --> 00:30:15
It's really a very nice kind
of thing to be able to do.

423
00:30:15 --> 00:30:19
OK, this works nicely.


424
00:30:19 --> 00:30:21
We'll get normal distributions.


425
00:30:21 --> 00:30:24
We can look at some things.


426
00:30:24 --> 00:30:26
There are two other kinds
of distributions I

427
00:30:26 --> 00:30:27
want to talk about.


428
00:30:27 --> 00:30:40
We can get rid of
this distraction.

429
00:30:40 --> 00:30:43
As you can imagine, I played a
lot with these things, just

430
00:30:43 --> 00:30:52
cause it was fun once I had it.


431
00:30:52 --> 00:30:57
You have these in your handout.


432
00:30:57 --> 00:31:01
So the one on the upper
right, is the Gaussian,

433
00:31:01 --> 00:31:07
or normal, distribution
we've been talking about.

434
00:31:07 --> 00:31:12
As I said earlier, quite
common, we see it a lot.

435
00:31:12 --> 00:31:17
The upper left is what's called
a, and these, by the way, all

436
00:31:17 --> 00:31:20
of these distributions are
symmetric, just in this

437
00:31:20 --> 00:31:23
particular picture.


438
00:31:23 --> 00:31:26
How do you spell symmetric,
one or two m's?

439
00:31:26 --> 00:31:29
I help here.


440
00:31:29 --> 00:31:30
That right?


441
00:31:30 --> 00:31:32
OK, thank you.


442
00:31:32 --> 00:31:35
And they're symmetric in the
sense that, if you take the

443
00:31:35 --> 00:31:40
mean, it looks the same on
both sides of the mean.

444
00:31:40 --> 00:31:43
Now in general, you
can have asymmetric

445
00:31:43 --> 00:31:48
distributions as well.


446
00:31:48 --> 00:31:51
But for simplicity, we'll
here look at symmetric ones.

447
00:31:51 --> 00:31:54
So we've seen the bell curve,
and then on the upper left is

448
00:31:54 --> 00:32:04
what's called the uniform.


449
00:32:04 --> 00:32:09
In a uniform distribution,
each value in the range

450
00:32:09 --> 00:32:15
is equally likely.


451
00:32:15 --> 00:32:18
So to characterize it,
you only need to give

452
00:32:18 --> 00:32:21
the range of values.


453
00:32:21 --> 00:32:26
I say the values range from
0 to 100, and it tells me

454
00:32:26 --> 00:32:30
everything I know about
the uniform distribution.

455
00:32:30 --> 00:32:35
Each value in that will occur
the same number of times.

456
00:32:35 --> 00:32:39
Have we seen a uniform
distribution?

457
00:32:39 --> 00:32:46
What have we seen
that's uniform here?

458
00:32:46 --> 00:32:46
Pardon?


459
00:32:46 --> 00:32:48
STUDENT: Playing dice.


460
00:32:48 --> 00:32:49
PROFESSOR: Playing dice.


461
00:32:49 --> 00:32:51
Exactly right.


462
00:32:51 --> 00:32:57
Each roll of the die
was equally likely.

463
00:32:57 --> 00:32:59
Between 1 and 6.


464
00:32:59 --> 00:33:03
So, we got a normal
distribution when I summed

465
00:33:03 --> 00:33:07
them, but if I gave you the
distribution of a single

466
00:33:07 --> 00:33:11
die, it would have
been uniform, right?

467
00:33:11 --> 00:33:14
So there's an interesting
lesson there.

468
00:33:14 --> 00:33:18
One die, the distribution was
uniform, but when I summed

469
00:33:18 --> 00:33:27
them, I ended up getting
a normal distribution.

470
00:33:27 --> 00:33:29
So where else do we see them?


471
00:33:29 --> 00:33:34
In principle, lottery winners
are uniformly distributed.

472
00:33:34 --> 00:33:38
Each number is equally
likely to come up.

473
00:33:38 --> 00:33:41
To a first approximation,
birthdays are uniformly

474
00:33:41 --> 00:33:44
distributed, things like that.


475
00:33:44 --> 00:33:49
But, in fact, they
rarely arise in nature.

476
00:33:49 --> 00:33:52
You'll hardly ever run a
physics experiment, or a

477
00:33:52 --> 00:33:55
biology experiment, or anything
like that, and come up with

478
00:33:55 --> 00:33:58
a uniform distribution.


479
00:33:58 --> 00:34:04
Nor do they arise very
often in complex systems.

480
00:34:04 --> 00:34:07
So if you look at what happens
in financial markets,

481
00:34:07 --> 00:34:10
none of the interesting
distributions are uniform.

482
00:34:10 --> 00:34:14
You know, the prices of stocks,
for example, are clearly

483
00:34:14 --> 00:34:16
not uniformly distributed.


484
00:34:16 --> 00:34:19
Up days and down days in
the stock market are not

485
00:34:19 --> 00:34:22
uniformly distributed.


486
00:34:22 --> 00:34:28
Winners of football games are
not uniformly distributed.

487
00:34:28 --> 00:34:32
People seem to like to use them
in games of chance, because

488
00:34:32 --> 00:34:35
they seem fair, but mostly
you see them only in

489
00:34:35 --> 00:34:39
invented things, rather
than real things.

490
00:34:39 --> 00:34:43
The third kind of distribution,
the one in the bottom, is the

491
00:34:43 --> 00:34:55
exponential distribution.


492
00:34:55 --> 00:34:59
That's actually quite
common in the real world.

493
00:34:59 --> 00:35:05
It's often used, for example,
to model arrival times.

494
00:35:05 --> 00:35:08
If you want to model the
frequency at which, say,

495
00:35:08 --> 00:35:16
automobiles arrive, get on the
Mass Turnpike, you would find

496
00:35:16 --> 00:35:20
that the arrivals
are exponential.

497
00:35:20 --> 00:35:25
We see with an exponential is,
things fall off much more

498
00:35:25 --> 00:35:34
steeply around the mean than
with the normal distribution.

499
00:35:34 --> 00:35:40
All right, that make sense?


500
00:35:40 --> 00:35:43
What else is exponentially
distributed?

501
00:35:43 --> 00:35:48
Requests for web pages
are often exponentially

502
00:35:48 --> 00:35:49
distributed.


503
00:35:49 --> 00:35:52
The amount of traffic
at a website.

504
00:35:52 --> 00:35:55
How frequently they arrive.


505
00:35:55 --> 00:35:59
We'll see much more starting
next week, or maybe even

506
00:35:59 --> 00:36:03
starting Thursday, about
exponential distributions, as

507
00:36:03 --> 00:36:06
we go on with a final case
study that we'll be dealing

508
00:36:06 --> 00:36:09
with in the course.


509
00:36:09 --> 00:36:16
You can think of each of these,
by the way, as increasing

510
00:36:16 --> 00:36:19
order of predictability.


511
00:36:19 --> 00:36:23
Uniform distribution means the
result is most unpredictable,

512
00:36:23 --> 00:36:27
it could be anything.


513
00:36:27 --> 00:36:34
A normal distribution says,
well, it's pretty predictable.

514
00:36:34 --> 00:36:37
Again, depending on the
standard deviation.

515
00:36:37 --> 00:36:44
If you guess the mean, you're
pretty close to right.

516
00:36:44 --> 00:36:47
The exponential is
very predictable.

517
00:36:47 --> 00:36:55
Most of the answers are
right around the mean.

518
00:36:55 --> 00:36:57
Now there are many other
distributions, there are Pareto

519
00:36:57 --> 00:37:01
distributions which have fat
tails, there are fractal

520
00:37:01 --> 00:37:05
distributions, there are
all sorts of things.

521
00:37:05 --> 00:37:09
We won't go into
to those details.

522
00:37:09 --> 00:37:13
Now, I hope you didn't find
this short excursion into

523
00:37:13 --> 00:37:17
statistics either too
boring or too confusing.

524
00:37:17 --> 00:37:20
The point was not to teach
you statistics, probability,

525
00:37:20 --> 00:37:24
we have multiple
courses to do that.

526
00:37:24 --> 00:37:26
But to give you some tools
that would help improve

527
00:37:26 --> 00:37:31
your intuition in
thinking about data.

528
00:37:31 --> 00:37:36
In closing, I want to
give a few words about

529
00:37:36 --> 00:37:38
the misuse of data.


530
00:37:38 --> 00:37:43
Since I think we misuse
data an awful lot.

531
00:37:43 --> 00:37:57
So, point number 0, as in the
most important, is beware of

532
00:37:57 --> 00:38:25
people who give you properties
of data, but not the data.

533
00:38:25 --> 00:38:29
We see that sort of
thing all the time.

534
00:38:29 --> 00:38:34
Where people come in, and they
say, OK, here it is, here's the

535
00:38:34 --> 00:38:37
mean value of the quiz, and
here's the standard deviation

536
00:38:37 --> 00:38:42
of the quiz, and that just
doesn't really tell you where

537
00:38:42 --> 00:38:44
you stand, in some sense.


538
00:38:44 --> 00:38:47
Because it's probably not
normally distributed.

539
00:38:47 --> 00:38:50
You want to see the data.


540
00:38:50 --> 00:38:53
At the very least, if you see
the data, you can then say,

541
00:38:53 --> 00:38:57
yeah, it is normally
distributed, so the standard

542
00:38:57 --> 00:39:01
deviation is meaningful,
or not meaningful.

543
00:39:01 --> 00:39:14
So, whenever you can, try and
get, at least, to see the data.

544
00:39:14 --> 00:39:18
So that's 1, or 0.


545
00:39:18 --> 00:39:24
1 is, well, all right.


546
00:39:24 --> 00:39:38
I'm going to test your Latin.


547
00:39:38 --> 00:39:41
Cum hoc ergo propter hoc.


548
00:39:41 --> 00:39:43
All right.


549
00:39:43 --> 00:39:49
I need a Latin scholar
to translate this.

550
00:39:49 --> 00:39:53
Did not one of you take
Latin in high school?

551
00:39:53 --> 00:39:55
We have someone who did.


552
00:39:55 --> 00:39:55
Go ahead.


553
00:39:55 --> 00:39:59
STUDENT: I think it means,
with this, therefore,

554
00:39:59 --> 00:40:01
because of this.


555
00:40:01 --> 00:40:03
PROFESSOR: Exactly right.


556
00:40:03 --> 00:40:08
With this, therefore,
because of this.

557
00:40:08 --> 00:40:09
I'm glad that at least
one person has a

558
00:40:09 --> 00:40:12
classical education.


559
00:40:12 --> 00:40:16
I don't, by the way.


560
00:40:16 --> 00:40:21
Essentially what this is
telling us, is that correlation

561
00:40:21 --> 00:40:39
does not imply causation.


562
00:40:39 --> 00:40:42
So sometimes two
things go together.

563
00:40:42 --> 00:40:45
They both go up,
they both go down.

564
00:40:45 --> 00:40:47
And people jump to the
conclusion that one

565
00:40:47 --> 00:40:49
causes the other.


566
00:40:49 --> 00:40:53
That there's a cause and
effect relationship.

567
00:40:53 --> 00:40:57
That is just not true.


568
00:40:57 --> 00:41:01
It's what's called
a logical fallacy.

569
00:41:01 --> 00:41:04
So we see some
examples of this.

570
00:41:04 --> 00:41:05
And you can get
into big trouble.

571
00:41:05 --> 00:41:08
So here's a very
interesting one.

572
00:41:08 --> 00:41:11
There was a very widely
reported epidemiological study,

573
00:41:11 --> 00:41:14
that's a medical study where
you get statistics about

574
00:41:14 --> 00:41:16
large populations.


575
00:41:16 --> 00:41:20
And it showed that women, who
are taking hormone replacement

576
00:41:20 --> 00:41:24
therapy, were found to have a
lower incidence of coronary

577
00:41:24 --> 00:41:28
heart disease than
women who didn't.

578
00:41:28 --> 00:41:32
This was a big study
of a lot of women.

579
00:41:32 --> 00:41:38
This led doctors to propose
that hormone replacement

580
00:41:38 --> 00:41:41
therapy for middle aged women
was protective against

581
00:41:41 --> 00:41:44
coronary heart disease.


582
00:41:44 --> 00:41:49
And in fact, in response to
this, a large number of medical

583
00:41:49 --> 00:41:50
societies recommended this.


584
00:41:50 --> 00:41:55
And a large number of women
were given this therapy.

585
00:41:55 --> 00:42:00
Later, controlled trials showed
that in fact, hormone

586
00:42:00 --> 00:42:04
replacement therapy in women
caused a small and significant

587
00:42:04 --> 00:42:10
increase in coronary
heart disease.

588
00:42:10 --> 00:42:13
So they had taken the fact that
these were correlated, said one

589
00:42:13 --> 00:42:17
causes the other, made a
prescription, and it turned

590
00:42:17 --> 00:42:20
out to be the wrong one.


591
00:42:20 --> 00:42:25
Now, how could this be?


592
00:42:25 --> 00:42:27
How could this be?


593
00:42:27 --> 00:42:33
It turned out that the women in
the original study who were

594
00:42:33 --> 00:42:39
taking the hormone replacement
therapy, tended to be from a

595
00:42:39 --> 00:42:43
higher socioeconomic group
than those who didn't.

596
00:42:43 --> 00:42:46
Because the therapy was not
covered by insurance, so the

597
00:42:46 --> 00:42:48
women who took it were wealthy.


598
00:42:48 --> 00:42:52
Turns out wealthy people do a
lot of other things that are

599
00:42:52 --> 00:42:54
protective of their hearts.


600
00:42:54 --> 00:42:57
And, therefore, are in general
healthier than poor people.

601
00:42:57 --> 00:42:59
This is not a surprise.


602
00:42:59 --> 00:43:02
Rich people are healthier
than poor people.

603
00:43:02 --> 00:43:09
And so in fact, it was this
third variable that was

604
00:43:09 --> 00:43:13
actually the meaningful one.


605
00:43:13 --> 00:43:30
This is what is called in
statistics, a lurking variable.

606
00:43:30 --> 00:43:34
Both of the things they were
looking at in this study, who

607
00:43:34 --> 00:43:39
took the therapy, and who had a
heart disease, each of those

608
00:43:39 --> 00:43:42
was correlated with the
lurking variable of

609
00:43:42 --> 00:43:46
socioeconomic position.


610
00:43:46 --> 00:43:51
And so, in effect, there was no
cause and effect relationship.

611
00:43:51 --> 00:43:54
And once they did another
study, in which the lurking

612
00:43:54 --> 00:43:59
variable was controlled, and
they looked at heart disease

613
00:43:59 --> 00:44:02
among rich women separately
from poor women, with this

614
00:44:02 --> 00:44:07
therapy, they discovered
that therapy was not good.

615
00:44:07 --> 00:44:10
It was, in fact, harmful.


616
00:44:10 --> 00:44:16
So this is a very important
moral to remember.

617
00:44:16 --> 00:44:21
When you look at correlations,
don't assume cause and effect.

618
00:44:21 --> 00:44:25
And don't assume that there
isn't a lurking variable that

619
00:44:25 --> 00:44:29
really is the dominant factor.


620
00:44:29 --> 00:44:35
So that's one statistical, a
second statistical worry.

621
00:44:35 --> 00:44:48
Number 2 is, beware of what's
called, non-response bias.

622
00:44:48 --> 00:44:52
Which is another fancy way of
saying, non-representative

623
00:44:52 --> 00:45:05
samples.


624
00:45:05 --> 00:45:08
No one doing a study beyond
the trivial can sample

625
00:45:08 --> 00:45:11
everybody or everything.


626
00:45:11 --> 00:45:17
And only mind readers can be
sure of what they've missed.

627
00:45:17 --> 00:45:18
Unless, of course,
people choose to miss

628
00:45:18 --> 00:45:20
things on purpose.


629
00:45:20 --> 00:45:22
Which you also see.


630
00:45:22 --> 00:45:24
And that brings me to
my next anecdote.

631
00:45:24 --> 00:45:31
A former professor at the
University of Nebraska, who

632
00:45:31 --> 00:45:34
later headed a group called The
Family Research Institute,

633
00:45:34 --> 00:45:37
which some of you may have
heard about, claimed that gay

634
00:45:37 --> 00:45:43
men have an average life
expectancy of 43 years.

635
00:45:43 --> 00:45:46
And they did a study full
of statistics showing

636
00:45:46 --> 00:45:49
that this was the case.


637
00:45:49 --> 00:45:53
And the key was, they
calculated the figure by

638
00:45:53 --> 00:45:57
checking gay newspapers
for obituaries and news

639
00:45:57 --> 00:46:00
about stories of death.


640
00:46:00 --> 00:46:04
So they went through the gay
newspapers, took a list of

641
00:46:04 --> 00:46:08
everybody whose obituary
appeared, how old they were

642
00:46:08 --> 00:46:13
when they died, took the
average, and said it was 43.

643
00:46:13 --> 00:46:15
Then they did a bunch of
statistics, with all sorts of

644
00:46:15 --> 00:46:19
tests, showing how, you know,
what the curves look like, the

645
00:46:19 --> 00:46:21
distributions, and
the significance.

646
00:46:21 --> 00:46:25
All the math was valid.


647
00:46:25 --> 00:46:28
The problem was, it was a very
unrepresentative sample.

648
00:46:28 --> 00:46:30
What was the most
unrepresentative

649
00:46:30 --> 00:46:32
thing about it?


650
00:46:32 --> 00:46:33
Somebody?


651
00:46:33 --> 00:46:38
STUDENT: Not all deaths
have obituaries.

652
00:46:38 --> 00:46:40
PROFESSOR: Well,
that's one thing.

653
00:46:40 --> 00:46:41
That's certainly true.


654
00:46:41 --> 00:46:44
But what else?


655
00:46:44 --> 00:46:48
Well, not all gay people
are dead, right?

656
00:46:48 --> 00:46:51
So if you're looking at
obituaries, you're in fact only

657
00:46:51 --> 00:46:53
getting -- I'm sure that's what
you were planning

658
00:46:53 --> 00:46:56
to say -- sorry.


659
00:46:56 --> 00:46:59
You're only getting the people
who are dead, so it's clearly

660
00:46:59 --> 00:47:02
going to make the number
look smaller, right?

661
00:47:02 --> 00:47:05
Furthermore, you're only
getting the ones that were

662
00:47:05 --> 00:47:10
reported in newspapers, the
newspapers are typically urban,

663
00:47:10 --> 00:47:18
rather than out in rural areas,
so it turns out, it's also

664
00:47:18 --> 00:47:23
biased against gays who chose
not come out of the closet, and

665
00:47:23 --> 00:47:26
therefore didn't
appear in these.

666
00:47:26 --> 00:47:29
Lots and lots of things
with the problems.

667
00:47:29 --> 00:47:31
Believe it or not, this
paper was published in

668
00:47:31 --> 00:47:34
a reputable journal.


669
00:47:34 --> 00:47:38
And someone checked all the
math, but missed the fact that

670
00:47:38 --> 00:47:49
all of that was irrelevant
because the sample was wrong.

671
00:47:49 --> 00:47:50
Data enhancement.


672
00:47:50 --> 00:47:53
It even sounds bad, right?


673
00:47:53 --> 00:47:56
You run an experiment, you get
your data, and you enhance it.

674
00:47:56 --> 00:47:59
It's kind of like when you ran
those physics experiments in

675
00:47:59 --> 00:48:02
high school, and you've got
answers that you knew didn't

676
00:48:02 --> 00:48:05
match the theory, so
you fudged the data?

677
00:48:05 --> 00:48:08
I know none of you would have
ever done that, but some

678
00:48:08 --> 00:48:09
people been known to do.


679
00:48:09 --> 00:48:12
That's not actually
what this means.

680
00:48:12 --> 00:48:16
What this means is, reading
more into the data than

681
00:48:16 --> 00:48:19
it actually implies.


682
00:48:19 --> 00:48:23
So well-meaning people are
often the guiltiest here.

683
00:48:23 --> 00:48:26
So here's another one
of my favorites.

684
00:48:26 --> 00:48:29
For example, there are
people who try to scare

685
00:48:29 --> 00:48:31
us into driving safely.


686
00:48:31 --> 00:48:33
Driving safely is a good thing.


687
00:48:33 --> 00:48:36
By telling holiday deaths.


688
00:48:36 --> 00:48:39
So you'll read things like,
400 killed on the highways

689
00:48:39 --> 00:48:42
over long weekend.


690
00:48:42 --> 00:48:47
It sounds really bad, until you
observe the fact that roughly

691
00:48:47 --> 00:48:51
400 people are killed
on any 3-day period.

692
00:48:51 --> 00:48:56
And in fact, it's no higher
on the holiday weekends.

693
00:48:56 --> 00:48:58
I'll bet you all thought
more people got killed

694
00:48:58 --> 00:48:59
on holiday weekends.


695
00:48:59 --> 00:49:02
Well, typically not.


696
00:49:02 --> 00:49:05
They just report how many died,
but they don't tell you the

697
00:49:05 --> 00:49:12
context, say, oh, by the
way, take any 3-day period.

698
00:49:12 --> 00:49:16
So the moral there is,
you really want to place

699
00:49:16 --> 00:49:20
the data in context.


700
00:49:20 --> 00:49:24
Data taken out of context
without comparison is

701
00:49:24 --> 00:49:27
usually meaningless.


702
00:49:27 --> 00:49:39
Another variance of
this is extrapolation.

703
00:49:39 --> 00:49:42
A commonly quoted statistic.


704
00:49:42 --> 00:49:45
Most auto accidents happen
within 10 miles of home.

705
00:49:45 --> 00:49:48
Anyone here heard
that statistic?

706
00:49:48 --> 00:49:53
It's true, but what
does it mean?

707
00:49:53 --> 00:49:55
Well, people tend to say it
means, it's dangerous to

708
00:49:55 --> 00:49:57
drive when you're near home.


709
00:49:57 --> 00:50:02
But in fact, most driving is
done within 10 miles of home.

710
00:50:02 --> 00:50:06
Furthermore, we don't
actually know where home is.

711
00:50:06 --> 00:50:09
Home is where the car is
supposedly garaged on the

712
00:50:09 --> 00:50:12
state registration forms.


713
00:50:12 --> 00:50:15
So, data enhancements would
suggest that I should

714
00:50:15 --> 00:50:18
register my car in Alaska.


715
00:50:18 --> 00:50:20
And then I would never be
driving within 10 miles of

716
00:50:20 --> 00:50:24
home, and I would
be much safer.

717
00:50:24 --> 00:50:32
Well, it's probably not a fact.


718
00:50:32 --> 00:50:37
So there are all sorts
of things on that.

719
00:50:37 --> 00:50:39
Well, I think I will come back
to this, because I have a

720
00:50:39 --> 00:50:42
couple more good stories which
I hate not to give you.

721
00:50:42 --> 00:50:45
So we'll come back on Thursday
and look at a couple of

722
00:50:45 --> 00:50:48
more things that can go
wrong with statistics.

723
00:50:48 --> 00:50:48