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PROFESSOR: Good morning.

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I want to start today's lecture
with, I guess one

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could call it, a confession.

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This isn't a real
6.00 lecture.

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00:00:34,650 --> 00:00:39,390
I'm standing here on June 30th
in an empty classroom dressed,

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for some bizarre reason,
as if it's the winter.

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I guess it's what the video
folks would call continuity.

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00:00:47,070 --> 00:00:51,110
What happened is Professor
Grimson gave a beautiful

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00:00:51,110 --> 00:00:58,350
lecture 13, for which we have a
lovely picture and no sound.

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We decided that probably those
of you watching this in

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00:01:01,930 --> 00:01:05,360
OpenCourseWare would be
unhappy just to watch

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Professor Grimson and not hear
him, so we're now re-taping

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the lecture.

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This is, actually, one of two
lectures we're going to be

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re-taping because we
had video problems.

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So when I say something
hilariously funny, and there

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is no laughter in
the room, it's

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because the room is empty.

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00:01:23,860 --> 00:01:28,183
Do me a favor and laugh out
there in video-land, so maybe

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at least someone will respond.

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OK, the last lecture closed with
a slight mystery, not a

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great mystery, but
a little mystery.

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We ran our simulation of the
drunkard's walk and got a

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result that wasn't credible.

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How did we know it
wasn't credible?

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Well we had worked out some
details on the blackboard of

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00:01:52,840 --> 00:01:56,250
what we thought would happen
with a small number of steps,

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and the results we got
didn't match that.

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00:01:59,130 --> 00:02:01,530
That told us we had
something wrong.

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00:02:01,530 --> 00:02:04,200
We asked you to go think about
it and come back today

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00:02:04,200 --> 00:02:06,840
prepared to tell us
what was wrong.

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00:02:06,840 --> 00:02:09,550
Well since there is no one here
to ask, I'm not going to

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ask the class to fix it, but
instead I have had the

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laborious task of fixing
it myself.

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00:02:17,390 --> 00:02:18,790
So let's look at it.

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The problem was in SimWalk.

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00:02:21,440 --> 00:02:24,140
And what we had is we had
the wrong argument here.

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We had numTrials Instead
of numSteps.

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And so it didn't
make any sense.

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I've now fixed it, and I,
now, want to run it.

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Well clearly, what I should do
to begin with, is run it on

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some examples for which we
already know the answer.

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00:02:41,970 --> 00:02:46,410
So I'll change drunkTest to
instead of running it on large

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numbers of steps, run
it on just a few.

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And let's see what we get.

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We run a lot of trials here,
since it's a short trial.

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00:03:03,950 --> 00:03:09,540
And what we see is for 100
trials of 0 steps, the mean

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was 0, the max was
0, the min was 0.

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Well that's exactly what
we saw when we looked

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at it on the board.

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It should happen.

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And the same thing with 1.

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Everything worked the way
it was supposed to work.

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It doesn't tell us the
program is perfect.

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It does tell us it works in at
least two examples, which was

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better than we had last time.

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All right now let's look at it
on a larger set of examples.

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I think I won't run 100 trials
here, because it'll take a

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little longer than we want.

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So let's run maybe 20.

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Let's see what we get.

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Well it's running
through here.

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Now we're getting examples that
seem much more credible.

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When we take 10 steps, the mean
is 2.85, and the max is

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between 6 and 1, or the max
is 6, the min is 1.

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It's what we would hope
for, that there's

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some dispersion there.

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And as we get up higher, we see
that for a 100,000 steps,

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the mean is 248, and there's
quite a spread between the max

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and the min.

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Finally we get to look at the
question that had us writing

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this code in the first place--

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how far should we expect this
drunk to be, given a

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particular amount of time.

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Well we can look at these
numbers and try and think

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about them in our heads, but in
fact, it's a lot easier to

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look at a picture.

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And this will get us to a theme
that we'll be getting to

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shortly in the course of how
do we visualize data.

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So here we have a
visualization.

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What I'm plotting, here, is the
mean number of the mean

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distance against the number of
steps for this random walk.

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And what we can see is, as we
knew, at 0, it's 0, and then

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it sort of goes up.

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And I've taken it only
up to a 1,000.

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And at 1,000 we're somewhere,
it looks

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like, around 25 steps.

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So we can learn something.

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00:05:30,750 --> 00:05:33,620
Well what can we learn
from this?

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How fast is it growing?

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Well it seems to grow
pretty fast, and

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then it flattens out.

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It looks to me like, roughly
speaking, the distance is

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growing as sort of close
to the square root of

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the number of steps.

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Well I could look at
it more closely.

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00:05:55,530 --> 00:05:59,640
We, actually, know this is not
exactly the square root.

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00:05:59,640 --> 00:06:04,740
Right, the square root of a
1,000 is not going to be 25,

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but I'm not going to delve
into the details of that.

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It's, actually, a little
bit complicated.

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We could derive it, but we
won't, because I want to get

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to what I think is a more
important message.

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How much should we infer from
this or from the numbers I

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displayed before, and I want
to say not too much.

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Because what we saw, if we go
look at what we had before, is

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quite a dispersion between
the max and the min.

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And furthermore, if we run it
again, the 20 trials, we're

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getting different answers.

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So what you can see, here, is
for 10,000 steps, here, my

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mean was 78, and here, for
10,000 steps, my mean is 90.

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00:07:18,450 --> 00:07:20,340
Here, the mean was 279.

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Here, it was 248.

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The maxs and the mins
are different.

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So I don't want to read too
much into that graph.

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Now one of the issues we should
have asked is, well

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it's the mean of how
many trials.

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I didn't tell you.

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And to be honest, I don't
quite remember.

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I think it was 20.

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But I don't have enough
information to interpret it.

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I need a lot more and that's
going to be what this whole

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next unit of the course is about
is how do we think about

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the results of programs when
the programs themselves are

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stochastic.

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00:08:04,990 --> 00:08:09,430
And this is important because,
as you will see, not only in

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this course, but as you progress
in your careers, if

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you're involved in engineering
or science, is that almost

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00:08:15,980 --> 00:08:19,220
everything in the real
world is stochastic.

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And in order to think about it,
we have to really think

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00:08:21,600 --> 00:08:25,360
pretty hard about what
those things mean.

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So I want to, now, pull back
and talk a little bit, sort

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00:08:28,920 --> 00:08:32,740
of, philosophically about
that, about the role of

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00:08:32,740 --> 00:08:33,990
randomness in computation--

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00:08:36,789 --> 00:08:40,140
something you probably haven't
seen a lot, if at all in the

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00:08:40,140 --> 00:08:42,760
other courses you've taken
if you're a freshman or

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00:08:42,760 --> 00:08:47,520
sophomore, and that's because
there's something really

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comforting about Newtonian
mechanics.

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When I first learned physics,
it was really comforting.

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00:08:54,350 --> 00:08:56,860
I learn the physics
of Isaac Newton.

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00:08:56,860 --> 00:08:59,410
You push down on one end of the
lever, the other end of

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00:08:59,410 --> 00:09:01,290
the lever goes up.

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00:09:01,290 --> 00:09:04,340
You throw a ball up in the air,
it travels a parabolic

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00:09:04,340 --> 00:09:05,590
path and lands.

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00:09:08,010 --> 00:09:13,310
F equals MA, that wonderful
rule of physics.

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Everything happened
for a reason.

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It was predictable.

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It was a great comfort to
some of us about it.

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And for centuries, that's the
way the world thought, from

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almost the beginning of the
times of science to, really,

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00:09:32,880 --> 00:09:35,640
if you look at history of
science, to today, people

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00:09:35,640 --> 00:09:38,400
believed the world was
deterministic.

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00:09:38,400 --> 00:09:40,720
And they liked it.

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00:09:40,720 --> 00:09:46,060
Then along came the so-called
Copenhagen Doctrine and

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quantum physics in the 20th
century, and the comforting

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00:09:51,790 --> 00:09:56,090
world of Newtonian physics
disappeared.

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00:09:56,090 --> 00:09:59,220
The Doctrine, the Copenhagen
Doctrine, led by the

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00:09:59,220 --> 00:10:04,110
physicists Bohr and Heisenberg,
argued that at its

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00:10:04,110 --> 00:10:08,780
most predictable and most
fundamental level, the

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00:10:08,780 --> 00:10:14,230
behavior of the physical world
cannot be predicted.

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00:10:14,230 --> 00:10:18,370
One can make probabilistic
statements of the form x is

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00:10:18,370 --> 00:10:23,070
highly likely to occur, but not
statements of the form x

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00:10:23,070 --> 00:10:26,360
is certain to occur--

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00:10:26,360 --> 00:10:28,850
period.

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00:10:28,850 --> 00:10:31,750
I can make the probabilistic
statement that if I take this

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00:10:31,750 --> 00:10:33,350
laser pointer and put
it on the table,

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00:10:33,350 --> 00:10:35,540
it won't fall through.

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00:10:35,540 --> 00:10:38,230
The molecules won't separate
in such a nice way that it

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00:10:38,230 --> 00:10:43,270
just drops through, but I can't
promise it won't happen.

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00:10:43,270 --> 00:10:46,590
Truth is I'm going to make that
promise anyway, because I

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00:10:46,590 --> 00:10:50,740
believe in probability, and it's
highly, highly probable.

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00:10:50,740 --> 00:10:55,100
But they tried to say that,
in fact, the world is all

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00:10:55,100 --> 00:10:55,870
stochastic.

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00:10:55,870 --> 00:10:59,330
Everything is probabilistic.

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00:10:59,330 --> 00:11:04,170
Other distinguished physicists
at the time, most notably

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00:11:04,170 --> 00:11:09,030
Einstein and Schrodinger,
vehemently disagreed.

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00:11:09,030 --> 00:11:13,090
This debate, it's hard to
believe, actually, roiled the

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00:11:13,090 --> 00:11:17,650
world of physics, philosophy,
and religion.

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00:11:17,650 --> 00:11:21,870
The heart of the debate was
the validity of something

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00:11:21,870 --> 00:11:23,550
called causal non-determinism.

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00:11:40,710 --> 00:11:42,410
The idea here--

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00:11:42,410 --> 00:11:47,010
causal means caused by
previous events.

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00:11:47,010 --> 00:11:51,140
So causal non-determinism was
the belief that not every

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00:11:51,140 --> 00:11:56,110
event is caused by
previous events.

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00:11:56,110 --> 00:12:00,050
Einstein and Schrodinger found
this view philosophically

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00:12:00,050 --> 00:12:05,720
unacceptable, as exemplified
by Einstein's often quoted

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00:12:05,720 --> 00:12:11,440
comment, "God does
not play dice."

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00:12:11,440 --> 00:12:15,950
What they argued is for
something called predictive

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00:12:15,950 --> 00:12:17,200
non-determinism.

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00:12:32,260 --> 00:12:37,510
The concept here was that our
inability to make accurate

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00:12:37,510 --> 00:12:40,880
measurements about the physical
world makes it

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00:12:40,880 --> 00:12:44,050
impossible to make precise
predictions about the future.

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00:12:46,550 --> 00:12:50,700
So this distinction was nicely
summed up again by Einstein,

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00:12:50,700 --> 00:12:54,410
who said, and I quote, "the
essentially statistical

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00:12:54,410 --> 00:12:59,060
character of contemporary
theory is solely to be

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00:12:59,060 --> 00:13:02,290
ascribed to the fact that this
theory operates with an

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00:13:02,290 --> 00:13:07,960
incomplete description of
physical systems," i.e.,

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00:13:07,960 --> 00:13:11,370
things are not unpredictable,
they just look unpredictable

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00:13:11,370 --> 00:13:16,330
because we don't know enough
about the initial states.

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00:13:16,330 --> 00:13:20,720
This question is still
unsettled in science.

207
00:13:20,720 --> 00:13:22,990
The good news is it probably
doesn't matter at all

208
00:13:22,990 --> 00:13:25,510
what the truth is.

209
00:13:25,510 --> 00:13:29,280
However you want to look at it,
we have to assume that the

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00:13:29,280 --> 00:13:33,410
world is non-deterministic,
because we can't, actually,

211
00:13:33,410 --> 00:13:35,290
predict it.

212
00:13:35,290 --> 00:13:37,810
So there's a little experiment
I sometimes do.

213
00:13:37,810 --> 00:13:39,665
I'll pretend to do it,
even though there

214
00:13:39,665 --> 00:13:41,350
are no students here.

215
00:13:41,350 --> 00:13:43,110
I take three coins--

216
00:13:43,110 --> 00:13:45,630
see if there are students in the
class, I ask the students

217
00:13:45,630 --> 00:13:48,370
to give me coins, and then
I try to steal them.

218
00:13:48,370 --> 00:13:50,200
But since there are no students,
I'm going to have to

219
00:13:50,200 --> 00:13:56,030
use my own coins, and I'm going
to ask, is at least one

220
00:13:56,030 --> 00:13:57,280
of them heads.

221
00:13:59,240 --> 00:14:05,250
Well the truth is, it's
completely predictable.

222
00:14:05,250 --> 00:14:07,470
I know the answer.

223
00:14:07,470 --> 00:14:09,030
But you don't know the
answer, because you

224
00:14:09,030 --> 00:14:11,320
can't see these coins.

225
00:14:11,320 --> 00:14:15,410
And so you might as well assume
it's probabilistic, and

226
00:14:15,410 --> 00:14:19,930
guess well, he put 3 coins down
at random, the odds of

227
00:14:19,930 --> 00:14:23,430
each one being a head is
1/2, so probably 1

228
00:14:23,430 --> 00:14:25,630
of the 3 is a head.

229
00:14:25,630 --> 00:14:29,620
And you're relying on
probability, because you don't

230
00:14:29,620 --> 00:14:32,200
know what's going on--

231
00:14:32,200 --> 00:14:33,450
predictive non-determinism.

232
00:14:36,720 --> 00:14:39,520
OK, and that's the way we're
going to deal with a lot of

233
00:14:39,520 --> 00:14:42,360
things going on for the
rest of the semester.

234
00:14:42,360 --> 00:14:46,410
We'll look at a lot of examples
where we have to act

235
00:14:46,410 --> 00:14:49,380
as if things are
non-deterministic.

236
00:14:49,380 --> 00:14:53,120
And that gets us to this notion
of what mathematicians

237
00:14:53,120 --> 00:14:54,580
call stochastic processes.

238
00:15:04,760 --> 00:15:10,090
A process is stochastic if it's
next state depends on

239
00:15:10,090 --> 00:15:14,910
both the previous states and
some random element.

240
00:15:48,550 --> 00:15:53,390
So now what I'm going to do is
pick up one of these coins,

241
00:15:53,390 --> 00:16:00,320
flip it in the air, put it down,
and ask, again, about

242
00:16:00,320 --> 00:16:04,270
the state of these
three coins.

243
00:16:04,270 --> 00:16:08,160
It depends upon the previous
state, because these two coins

244
00:16:08,160 --> 00:16:11,880
have the same value they had in
the previous state, plus a

245
00:16:11,880 --> 00:16:14,720
stochastic element, a
probabilistic element, a

246
00:16:14,720 --> 00:16:16,340
random element--

247
00:16:16,340 --> 00:16:18,680
the value of that coin,
which I just flipped.

248
00:16:21,320 --> 00:16:26,520
OK, most programming languages,
including Python,

249
00:16:26,520 --> 00:16:31,450
include simple ways to write
programs that use randomness.

250
00:16:31,450 --> 00:16:38,110
As we'll see, as we've already
seen in Python with our

251
00:16:38,110 --> 00:16:47,930
drunkard's walk, we use the
function random.choice, which,

252
00:16:47,930 --> 00:16:52,020
given a set of values
at random, chose

253
00:16:52,020 --> 00:16:54,460
one of those values.

254
00:16:54,460 --> 00:16:58,900
That function and almost all
of the other functions in

255
00:16:58,900 --> 00:17:04,380
Python that involve randomness
are implemented using

256
00:17:04,380 --> 00:17:05,630
something called
random.random.

257
00:17:10,200 --> 00:17:14,579
This function generates a random
float that's greater

258
00:17:14,579 --> 00:17:23,119
than 0 and no greater than 1.0,
So you get one of the

259
00:17:23,119 --> 00:17:26,050
infinite or seemingly infinite
number of floating point

260
00:17:26,050 --> 00:17:31,890
values that are greater than
0 and no greater than 1.

261
00:17:31,890 --> 00:17:35,360
So let's go look at
another example of

262
00:17:35,360 --> 00:17:36,610
the stochastic process.

263
00:17:40,510 --> 00:17:41,885
We're going to look
at throwing dice.

264
00:17:48,170 --> 00:17:55,350
So I've got something called
rollDie, which chooses a value

265
00:17:55,350 --> 00:17:57,630
between 1 and 6.

266
00:17:57,630 --> 00:17:59,790
For those of you who have
never gambled with

267
00:17:59,790 --> 00:18:01,820
dice, it's a cube.

268
00:18:01,820 --> 00:18:04,410
You roll it, and it has a value
between 1 and 6 that

269
00:18:04,410 --> 00:18:06,620
shows up at random.

270
00:18:06,620 --> 00:18:12,760
And then I've got this little
program called testRoll that

271
00:18:12,760 --> 00:18:17,420
rolls a bunch of dice and
comes up with an answer.

272
00:18:17,420 --> 00:18:21,190
All right, so let's
see what happens.

273
00:18:24,000 --> 00:18:27,060
Actually, before we do that,
let me ask you--

274
00:18:27,060 --> 00:18:29,550
we can look at a question.

275
00:18:29,550 --> 00:18:34,350
Imagine I roll it, and
I run it some large

276
00:18:34,350 --> 00:18:37,900
number of times, 10.

277
00:18:37,900 --> 00:18:40,940
Would you expect to
see the value--

278
00:18:44,890 --> 00:18:50,780
more likely see that value or
might more likely see a value

279
00:18:50,780 --> 00:18:52,030
that looks like this.

280
00:19:02,120 --> 00:19:05,390
Which of these values is more
likely to come up from my

281
00:19:05,390 --> 00:19:07,870
random rolls of the die?

282
00:19:12,380 --> 00:19:14,860
Well when I take a vote--

283
00:19:14,860 --> 00:19:16,230
if I take a vote--

284
00:19:16,230 --> 00:19:21,110
historically in this class, this
strikes people as more

285
00:19:21,110 --> 00:19:23,840
likely to happen than this.

286
00:19:23,840 --> 00:19:27,790
But it's a trick question,
because as it happens, they're

287
00:19:27,790 --> 00:19:29,920
equally likely.

288
00:19:29,920 --> 00:19:36,590
And the reason they're equally
likely is each roll is

289
00:19:36,590 --> 00:19:40,120
independent of the
previous rolls.

290
00:19:43,240 --> 00:19:46,620
And as we'll see in our
excursion in probability and

291
00:19:46,620 --> 00:19:53,030
randomness, independence is a
very important assumption.

292
00:19:53,030 --> 00:19:59,020
In a stochastic process, two
events are independent if the

293
00:19:59,020 --> 00:20:02,700
outcome of one event has
no influence on the

294
00:20:02,700 --> 00:20:05,540
outcome of the other.

295
00:20:05,540 --> 00:20:10,490
The events are independent if
the outcome of one event has

296
00:20:10,490 --> 00:20:14,870
no influence on the outcome
of the other.

297
00:20:14,870 --> 00:20:17,880
So it's a bit easier to think
about this, maybe, if we

298
00:20:17,880 --> 00:20:21,050
simplify the situation for the
moment to think about flipping

299
00:20:21,050 --> 00:20:25,830
coins, which have either heads
or tails, and I'll look at the

300
00:20:25,830 --> 00:20:30,590
value 0 or 1 as the
examples there.

301
00:20:30,590 --> 00:20:32,240
That means I can use--

302
00:20:32,240 --> 00:20:34,960
I have a binary die
for some reason.

303
00:20:34,960 --> 00:20:41,380
So as we've seen before, if I
flip a coin ten times, how

304
00:20:41,380 --> 00:20:45,330
many different possibilities,
sequences

305
00:20:45,330 --> 00:20:47,220
of 0 and 1 are there?

306
00:20:47,220 --> 00:20:49,730
Well we've seen this kind of
thing a lot of times already

307
00:20:49,730 --> 00:20:51,880
this semester.

308
00:20:51,880 --> 00:20:55,930
There are 2 to the 10 binary
numbers of 10 digits, so we

309
00:20:55,930 --> 00:20:57,590
know that there are 2 to
the 10 possibilities.

310
00:21:00,870 --> 00:21:02,210
Each of these 2 to the 10

311
00:21:02,210 --> 00:21:06,350
possibilities are equally likely.

312
00:21:06,350 --> 00:21:13,140
So the number in which I have
all 0's is no more likely than

313
00:21:13,140 --> 00:21:16,670
the number of all 1's, is
no more likely than some

314
00:21:16,670 --> 00:21:19,580
seemingly random combinations
of 0 and 1.

315
00:21:24,800 --> 00:21:28,640
So it's a very small
probability.

316
00:21:28,640 --> 00:21:33,370
So what's the probability
of getting all 1's?

317
00:21:33,370 --> 00:21:36,060
It's 1 out of 2 to the 10.

318
00:21:39,120 --> 00:21:41,580
What's the probability
of getting all 0's?

319
00:21:41,580 --> 00:21:43,850
1 out of 2 to the 10.

320
00:21:43,850 --> 00:21:46,880
What's the probability of any
combination you would happen

321
00:21:46,880 --> 00:21:49,070
to pick of 0's and 1's?

322
00:21:49,070 --> 00:21:50,970
1 over 2 to the 10.

323
00:21:54,310 --> 00:21:57,520
I know I'm belaboring this
point, but the point I want to

324
00:21:57,520 --> 00:22:02,830
make is that when we talk about
some result having a

325
00:22:02,830 --> 00:22:07,000
particular probability, we are
asking, essentially, the

326
00:22:07,000 --> 00:22:19,960
question, what fraction of the
possible results have the

327
00:22:19,960 --> 00:22:22,550
property we're testing for.

328
00:22:26,720 --> 00:22:30,120
So I ask what property
are all 1's.

329
00:22:30,120 --> 00:22:32,225
I'm saying what fraction
are all 1's.

330
00:22:32,225 --> 00:22:36,350
If I say well, the properties
are exactly four 1's.

331
00:22:36,350 --> 00:22:40,600
What fraction of these numbers
have exactly four 1's in them?

332
00:22:40,600 --> 00:22:42,000
Whatever I want.

333
00:22:42,000 --> 00:22:47,250
So probabilities will
always be fractions.

334
00:22:47,250 --> 00:22:51,180
That's important because it
means that when we talk about

335
00:22:51,180 --> 00:22:55,450
the probability of some event
occurring, we know it has to

336
00:22:55,450 --> 00:23:00,060
be somewhere between 0 and 1.

337
00:23:00,060 --> 00:23:03,420
Probabilities are never
less than 0.

338
00:23:03,420 --> 00:23:05,900
They're never more than 1.

339
00:23:05,900 --> 00:23:10,710
Cannot happen, guaranteed
to happen, usually

340
00:23:10,710 --> 00:23:11,960
somewhere in between.

341
00:23:14,740 --> 00:23:18,570
All right, that's the key
thing to remember when

342
00:23:18,570 --> 00:23:19,820
thinking about probabilities.

343
00:23:22,230 --> 00:23:25,980
Suppose I want to ask, what's
the probability of getting

344
00:23:25,980 --> 00:23:30,240
some sequence of coin flips
other than all 1's.

345
00:23:33,040 --> 00:23:37,990
Well I know the probability of
getting all 1's is 1 out of 2

346
00:23:37,990 --> 00:23:39,240
to the 10th.

347
00:23:41,400 --> 00:23:43,750
I know the probability
of getting some

348
00:23:43,750 --> 00:23:46,350
sequence of flips is 1.

349
00:23:46,350 --> 00:23:49,110
It's certain I'll get one
of those numbers.

350
00:23:49,110 --> 00:23:52,610
So the answer is the probability
of not getting all

351
00:23:52,610 --> 00:23:55,680
1's is 1 minus 1 over
2 to the 10th.

352
00:23:58,200 --> 00:24:02,490
This is an important
trick to remember.

353
00:24:02,490 --> 00:24:07,010
Typically we have two ways of
computing probabilities.

354
00:24:07,010 --> 00:24:11,240
We can either compute it
directly, as I did when I

355
00:24:11,240 --> 00:24:16,170
computed the probability of
getting all 1's, or we can

356
00:24:16,170 --> 00:24:22,940
compute the probability of
something not happening by

357
00:24:22,940 --> 00:24:26,450
subtracting one probability
from another,

358
00:24:26,450 --> 00:24:29,240
this 1 minus trick.

359
00:24:29,240 --> 00:24:32,720
And so you'll see me using
this formulation a lot of

360
00:24:32,720 --> 00:24:36,760
times, And we'll talk as we
go forward about when

361
00:24:36,760 --> 00:24:38,910
you do it which way.

362
00:24:38,910 --> 00:24:45,910
All right so let's go back,
finally, to our six-sided die.

363
00:24:45,910 --> 00:24:50,020
How many sequences are there
of length 10 for that?

364
00:24:50,020 --> 00:24:51,290
2 to the 10th?

365
00:24:51,290 --> 00:24:52,620
No.

366
00:24:52,620 --> 00:24:54,370
6 to the 10th.

367
00:24:54,370 --> 00:24:57,390
Because unlike the coin where
we only had 2 possibilities,

368
00:24:57,390 --> 00:25:00,020
we now have 6 possibilities.

369
00:25:00,020 --> 00:25:05,120
So there are 6 to the 10th
different sequences of rolls I

370
00:25:05,120 --> 00:25:08,470
could get, quite a few.

371
00:25:08,470 --> 00:25:14,600
So the probability of getting 10
consecutive 1's is 1 over 6

372
00:25:14,600 --> 00:25:17,070
to the 10th.

373
00:25:17,070 --> 00:25:19,640
And of course, the probability
of getting this sequence,

374
00:25:19,640 --> 00:25:23,320
here, is also 1 over 6 to
the 10th, so we see

375
00:25:23,320 --> 00:25:26,815
that they are equal.

376
00:25:26,815 --> 00:25:31,080
OK, we're going to spend a lot
more time on probability and

377
00:25:31,080 --> 00:25:34,155
randomized and stochastic
algorithms.

378
00:25:34,155 --> 00:25:38,850
But before I do that, I want to
take a brief digression and

379
00:25:38,850 --> 00:25:41,570
return to the topic we looked
at a little earlier this

380
00:25:41,570 --> 00:25:46,220
morning, which was data
visualization, plotting.

381
00:25:46,220 --> 00:25:48,610
I want to do this
for two reasons.

382
00:25:48,610 --> 00:25:51,360
One, it's really important.

383
00:25:51,360 --> 00:25:54,650
It's something that all of us
do a lot of in the course of

384
00:25:54,650 --> 00:25:59,430
our work, but it also will just
make it a lot easier for

385
00:25:59,430 --> 00:26:03,720
me to talk about probability
and stochastics when I can

386
00:26:03,720 --> 00:26:07,040
draw some pretty pictures to
illustrate what's going on.

387
00:26:10,450 --> 00:26:17,720
Now many people, most of us,
probably, when we're writing

388
00:26:17,720 --> 00:26:21,470
code to do something, focus on
writing programs that perform

389
00:26:21,470 --> 00:26:25,000
some complicated analysis
of the data,

390
00:26:25,000 --> 00:26:26,960
and then print something.

391
00:26:26,960 --> 00:26:31,220
We don't spend enough time, I
think, worrying about how the

392
00:26:31,220 --> 00:26:35,760
results of our analyses are
presented so that somebody

393
00:26:35,760 --> 00:26:38,460
else can make sense of them, or
in fact, we can understand

394
00:26:38,460 --> 00:26:40,730
them better ourselves.

395
00:26:40,730 --> 00:26:44,660
Sometimes text is the best way,
but sometimes there's a

396
00:26:44,660 --> 00:26:49,140
lot of truth to the Chinese
proverb that a picture's

397
00:26:49,140 --> 00:26:53,330
meaning can express
10,000 words.

398
00:26:53,330 --> 00:26:55,840
Now most of us, sort
of, believe this.

399
00:26:55,840 --> 00:26:57,480
Why don't we do it?

400
00:26:57,480 --> 00:26:59,930
Well because in most programming
languages, it's

401
00:26:59,930 --> 00:27:03,010
hard to draw pretty pictures.

402
00:27:03,010 --> 00:27:05,670
One of the reasons we use
Python in this class is

403
00:27:05,670 --> 00:27:09,770
because in Python, it's easy to
draw pretty pictures or, at

404
00:27:09,770 --> 00:27:12,530
least, to make plots.

405
00:27:12,530 --> 00:27:14,020
Why is that?

406
00:27:14,020 --> 00:27:16,560
It's because somebody--

407
00:27:16,560 --> 00:27:17,780
not me--

408
00:27:17,780 --> 00:27:19,500
went to the trouble of building

409
00:27:19,500 --> 00:27:20,750
something called PyLab.

410
00:27:26,170 --> 00:27:30,780
PyLab is a Python library that
provides many of the

411
00:27:30,780 --> 00:27:34,050
facilities of something
called MATLAB.

412
00:27:41,170 --> 00:27:47,260
If you're an MIT student, the
probability of your graduating

413
00:27:47,260 --> 00:27:51,740
without using MATLAB
is very low.

414
00:27:51,740 --> 00:27:54,980
It is something people
use a lot.

415
00:27:54,980 --> 00:27:57,330
It's not my favorite programming
language.

416
00:27:57,330 --> 00:27:59,560
It has its utility.

417
00:27:59,560 --> 00:28:02,590
I like Python, because
it brings--

418
00:28:02,590 --> 00:28:07,600
a lot of the features of MATLAB
are easy to use in a

419
00:28:07,600 --> 00:28:10,270
programming language that
I find much more

420
00:28:10,270 --> 00:28:12,170
convivial than MATLAB.

421
00:28:14,710 --> 00:28:17,830
All right I'm not going to give
you a complete tutorial

422
00:28:17,830 --> 00:28:20,030
for PyLab here.

423
00:28:20,030 --> 00:28:23,630
It would take a long time,
and it would be boring.

424
00:28:23,630 --> 00:28:26,960
Instead I'm going to give you
a few examples, and in fact,

425
00:28:26,960 --> 00:28:32,166
focus primarily on the plotting
capabilities.

426
00:28:35,810 --> 00:28:39,880
And the good news is the
plotting capabilities in PyLab

427
00:28:39,880 --> 00:28:43,760
are almost identical to those in
MATLAB, so if you learn how

428
00:28:43,760 --> 00:28:48,580
to do it here, you'll already
know how to do it here.

429
00:28:48,580 --> 00:28:54,046
For details you should
take a look at--

430
00:28:54,046 --> 00:28:56,060
let me make sure I write
this correctly--

431
00:28:59,140 --> 00:29:15,360
this website,
matplotlib.sourceforge.net,

432
00:29:15,360 --> 00:29:18,750
and it's a very nicely put
together website that will

433
00:29:18,750 --> 00:29:22,880
give you all of the capabilities
of plotting.

434
00:29:22,880 --> 00:29:27,000
Also you'll find in the class
website, I've written a little

435
00:29:27,000 --> 00:29:31,330
chapter of a book about how to
do this sort of thing, and you

436
00:29:31,330 --> 00:29:32,750
may also find that helpful.

437
00:29:35,510 --> 00:29:39,170
I should point out that PyLab
is not part of the standard

438
00:29:39,170 --> 00:29:41,240
Python distribution.

439
00:29:41,240 --> 00:29:44,610
It has to be installed on your
computer, and again, there are

440
00:29:44,610 --> 00:29:46,890
instructions about how
to do this posted

441
00:29:46,890 --> 00:29:49,730
on the class website.

442
00:29:49,730 --> 00:29:53,290
All right let's start with
something very simple.

443
00:29:58,400 --> 00:29:59,480
We'll look at that, here.

444
00:29:59,480 --> 00:30:02,930
I'm beginning by importing.

445
00:30:08,500 --> 00:30:09,700
I'll import PyLab.

446
00:30:09,700 --> 00:30:13,220
You have to import
it to use it.

447
00:30:13,220 --> 00:30:16,470
And then I'm going to
plot two things.

448
00:30:16,470 --> 00:30:19,560
So what I want to observe,
here, is I'm

449
00:30:19,560 --> 00:30:20,980
going to plot two vectors--

450
00:30:20,980 --> 00:30:26,160
the vector 1, 2, 3, 4 and
the vector 1, 2, 3, 4.

451
00:30:26,160 --> 00:30:27,830
These are the x-coordinates.

452
00:30:27,830 --> 00:30:29,850
These are the y-coordinates.

453
00:30:29,850 --> 00:30:33,540
So we'll get a two-dimensional
plot, x versus y.

454
00:30:33,540 --> 00:30:36,900
And it's very important that
these two vectors be of the

455
00:30:36,900 --> 00:30:38,150
same length.

456
00:30:40,420 --> 00:30:43,170
Doubtless when you're using this
in your problem sets, you

457
00:30:43,170 --> 00:30:46,310
will screw up and you'll get
an error message, which you

458
00:30:46,310 --> 00:30:48,940
will have a hard time
interpreting, but what it

459
00:30:48,940 --> 00:30:51,340
probably is going to boil down
to is you've done something

460
00:30:51,340 --> 00:30:53,540
wrong, and you're plotting
things that are

461
00:30:53,540 --> 00:30:56,700
not the same length.

462
00:30:56,700 --> 00:30:59,290
After I plot these two things,
I'm going to type

463
00:30:59,290 --> 00:31:03,340
"pyLab.show," which will
put the plots up

464
00:31:03,340 --> 00:31:04,760
for us to look at.

465
00:31:04,760 --> 00:31:06,320
So let's do that now.

466
00:31:16,640 --> 00:31:20,530
So the first plot in
our straight line--

467
00:31:20,530 --> 00:31:24,900
1 versus 1, 2 versus 2, 3
versus 3, 4 versus 4--

468
00:31:24,900 --> 00:31:26,110
got that.

469
00:31:26,110 --> 00:31:30,500
Then it plotted this rather
funny looking zigzag we saw,

470
00:31:30,500 --> 00:31:35,280
kind of just randomly chosen
in the other one.

471
00:31:35,280 --> 00:31:37,615
And the plots will always look
something like this.

472
00:31:40,730 --> 00:31:46,295
I should mention, if we go look
at the code, pyLab.show--

473
00:31:49,030 --> 00:31:53,190
if I haven't said that, the plot
would not have appeared

474
00:31:53,190 --> 00:31:54,950
on my screen.

475
00:31:54,950 --> 00:31:59,440
PyLab would have produced it,
but not displayed it.

476
00:31:59,440 --> 00:32:03,000
You may think that's silly,
but in fact it's useful,

477
00:32:03,000 --> 00:32:06,350
because most of the time when
I'm writing programs that do

478
00:32:06,350 --> 00:32:10,200
plotting, I don't want to
see them on my screen.

479
00:32:10,200 --> 00:32:12,550
I'm producing a whole bunch of
plots, and I'm going to write

480
00:32:12,550 --> 00:32:16,170
them to files that I will then
look at later or include in a

481
00:32:16,170 --> 00:32:19,940
paper or a lecture or something,
so it makes me say

482
00:32:19,940 --> 00:32:23,230
that I want to see it.

483
00:32:23,230 --> 00:32:28,280
I should point out, depending
upon your operating system, it

484
00:32:28,280 --> 00:32:32,460
can be pretty annoying, because
if you try and do this

485
00:32:32,460 --> 00:32:36,180
twice in your code, something
bad can happen.

486
00:32:36,180 --> 00:32:39,350
The code can hang.

487
00:32:39,350 --> 00:32:41,600
Therefore, you should
only execute

488
00:32:41,600 --> 00:32:45,010
pyLab.show once per program.

489
00:32:45,010 --> 00:32:48,510
And it should always be the
last thing you executed,

490
00:32:48,510 --> 00:32:52,140
because once you execute
pyLab.show, the program will

491
00:32:52,140 --> 00:32:54,960
stop running, essentially.

492
00:32:54,960 --> 00:32:55,730
It's annoying.

493
00:32:55,730 --> 00:32:58,280
I wish it weren't that
way, but it is.

494
00:32:58,280 --> 00:33:01,840
So live with it.

495
00:33:01,840 --> 00:33:06,095
All right let's go look back
at our graph, our plot.

496
00:33:11,900 --> 00:33:15,450
At the top is a title.

497
00:33:15,450 --> 00:33:17,080
This is the default title.

498
00:33:17,080 --> 00:33:19,520
It says figure 1.

499
00:33:19,520 --> 00:33:22,130
Later we'll see that I could
have given it a much better

500
00:33:22,130 --> 00:33:23,990
title than that.

501
00:33:23,990 --> 00:33:29,230
Then it's got the values of the
x and y-axes and a bunch

502
00:33:29,230 --> 00:33:34,050
of things down at the bottom
that we could point out.

503
00:33:34,050 --> 00:33:39,890
You can zoom in on the plots,
using that or zoom out.

504
00:33:39,890 --> 00:33:42,390
You can use this funny icon.

505
00:33:42,390 --> 00:33:46,940
It happens to be a floppy disk
icon, something that probably

506
00:33:46,940 --> 00:33:49,190
most of you have never seen.

507
00:33:49,190 --> 00:33:52,130
Congratulations if you've never
seen a floppy disk.

508
00:33:52,130 --> 00:33:54,750
Your life is better than
it would be had you had

509
00:33:54,750 --> 00:33:56,580
to deal with them.

510
00:33:56,580 --> 00:33:59,370
But that's used for saving
them to a file.

511
00:33:59,370 --> 00:34:00,890
You can move around in it.

512
00:34:00,890 --> 00:34:04,130
You can get back to what the
original figure was, a whole

513
00:34:04,130 --> 00:34:05,380
bunch of useful things.

514
00:34:05,380 --> 00:34:09,550
I suggest you just bring this
up and play with it.

515
00:34:14,520 --> 00:34:19,110
One of the things I should
mention here is you'll note

516
00:34:19,110 --> 00:34:23,639
that when I produced this,
I only use four points--

517
00:34:23,639 --> 00:34:26,800
1, 2, 3, and 4, say,
for this one.

518
00:34:26,800 --> 00:34:31,732
Yet it looks as if I have
a continuous plot.

519
00:34:31,732 --> 00:34:37,600
You know, it claims that
1.5 and 1.5 match.

520
00:34:37,600 --> 00:34:41,730
This can be very deceptive,
as we'll see later on, and

521
00:34:41,730 --> 00:34:45,150
probably it might have been
better for me to plot not

522
00:34:45,150 --> 00:34:48,960
lines here, but points,
indicating there are only four

523
00:34:48,960 --> 00:34:51,420
points in this graph.

524
00:34:51,420 --> 00:34:54,040
It's more apparent here, where
we see this funny looking

525
00:34:54,040 --> 00:34:58,070
zigzag, implying some
complicated relationship

526
00:34:58,070 --> 00:35:00,370
amongst the points,
which, actually,

527
00:35:00,370 --> 00:35:02,640
probably doesn't exist.

528
00:35:02,640 --> 00:35:09,060
And again later on we'll
see ways to avoid that.

529
00:35:09,060 --> 00:35:10,820
OK.

530
00:35:10,820 --> 00:35:14,940
It is, of course, possible to
produce more than one figure.

531
00:35:14,940 --> 00:35:16,190
So let's look at this.

532
00:35:18,720 --> 00:35:22,630
We'll comment this piece
out for now.

533
00:35:34,480 --> 00:35:37,900
And we'll run this code.

534
00:35:37,900 --> 00:35:41,540
So this says I want to plot
something on a figure I'm

535
00:35:41,540 --> 00:35:45,630
calling figure 1, as before,
And then I'm going to

536
00:35:45,630 --> 00:35:46,620
go to figure 2.

537
00:35:46,620 --> 00:35:49,650
So now instead of plotting
both of these on the same

538
00:35:49,650 --> 00:35:52,950
figure, I'm going to put them
on separate figures.

539
00:35:52,950 --> 00:35:59,100
And then pyLab.savefigure will,
actually, create a file

540
00:35:59,100 --> 00:36:02,880
in the directory in which I'm
running the program and save

541
00:36:02,880 --> 00:36:06,260
it-- called firstsaved And then
this will be secondsaved.

542
00:36:06,260 --> 00:36:07,640
And now I can run it.

543
00:36:14,820 --> 00:36:17,150
And now I have two figures--

544
00:36:17,150 --> 00:36:19,390
figure 1, as before--

545
00:36:19,390 --> 00:36:21,740
well not quite as before,
a different figure--

546
00:36:21,740 --> 00:36:22,990
and figure 2.

547
00:36:27,150 --> 00:36:30,850
Again, nothing very
magical there.

548
00:36:30,850 --> 00:36:33,530
And if we look in my directory,
we should see, I

549
00:36:33,530 --> 00:36:39,380
hope, firstsaved and
secondsaved, so if we look at

550
00:36:39,380 --> 00:36:44,770
secondsaved, we'll
see, there it is.

551
00:36:44,770 --> 00:36:48,640
And just to show that I'm not
cheating, you can notice that

552
00:36:48,640 --> 00:36:53,620
the time stamp is today at 10:34
AM, which happens to be

553
00:36:53,620 --> 00:36:58,170
when I'm giving this
particular lecture.

554
00:36:58,170 --> 00:37:03,025
All right you can put that
away now and continue.

555
00:37:08,000 --> 00:37:24,180
Now what I want you to notice
is the last one.

556
00:37:24,180 --> 00:37:27,030
I just gave it one argument--

557
00:37:27,030 --> 00:37:30,820
5, 6, 7 and 10.

558
00:37:30,820 --> 00:37:32,185
And if we look at what
it plotted--

559
00:37:41,100 --> 00:37:43,010
actually, I think I put that
in figure 1, didn't I?

560
00:37:47,110 --> 00:37:48,360
You'll see, here it is.

561
00:37:51,900 --> 00:37:57,480
And it's made up some values
for the x-axis.

562
00:37:57,480 --> 00:38:00,380
If you only give it one set of
values, it assumes it's the

563
00:38:00,380 --> 00:38:03,825
y-axis, and it finds values.

564
00:38:06,340 --> 00:38:10,890
All right now what values
is it going to

565
00:38:10,890 --> 00:38:13,090
choose for the x-axis?

566
00:38:13,090 --> 00:38:16,690
Well this is Python, so
surprise, surprise, the first

567
00:38:16,690 --> 00:38:21,610
value is 0, 1, 2, and 3.

568
00:38:21,610 --> 00:38:23,090
It's how I get the
four values.

569
00:38:25,890 --> 00:38:32,320
Now we could look at another
example, a slightly more

570
00:38:32,320 --> 00:38:33,570
interesting one.

571
00:38:37,190 --> 00:38:39,790
Comment this out, so we don't
look at the boring stuff over

572
00:38:39,790 --> 00:38:41,040
and over again.

573
00:38:46,330 --> 00:38:50,325
I've written a little program
to calculate interest.

574
00:38:58,270 --> 00:39:04,090
So I'm going to start with an
initial principal, here, of

575
00:39:04,090 --> 00:39:08,900
1,000, an interest rate
of 5%, 20 years, and

576
00:39:08,900 --> 00:39:12,490
just do compound interest.

577
00:39:12,490 --> 00:39:15,430
This, you all would know
how to write this.

578
00:39:15,430 --> 00:39:17,450
And I'm going to plot it
and see what we get.

579
00:39:26,350 --> 00:39:28,380
All right so what we have--

580
00:39:28,380 --> 00:39:32,040
something here, which, sort
of, shows the kind of

581
00:39:32,040 --> 00:39:36,420
beautiful growth you get with
compound interest, what the

582
00:39:36,420 --> 00:39:40,050
finance people call the magic
of compounding, which will

583
00:39:40,050 --> 00:39:43,710
make us all rich in principle,
until we see what the

584
00:39:43,710 --> 00:39:45,630
markets really do.

585
00:39:45,630 --> 00:39:48,790
But at any rate, for now we can
look at it, and it looks

586
00:39:48,790 --> 00:39:50,040
very pretty.

587
00:39:53,090 --> 00:39:56,850
But I don't know
what it means.

588
00:39:56,850 --> 00:40:00,560
I look and say, "oh, its figure
1." Well that's not

589
00:40:00,560 --> 00:40:05,670
very informative, and x goes
from 0 to 20, but if I haven't

590
00:40:05,670 --> 00:40:07,380
told you, you wouldn't
know what that meant.

591
00:40:07,380 --> 00:40:12,720
And y from 10,000 up to
28,000, but again, you

592
00:40:12,720 --> 00:40:13,970
wouldn't know what that means.

593
00:40:16,690 --> 00:40:19,340
We see this all the time.

594
00:40:19,340 --> 00:40:20,590
It's not a good thing.

595
00:40:24,340 --> 00:40:26,820
It's a bad thing, in fact.

596
00:40:26,820 --> 00:40:31,080
All plots should have
informative titles, and all

597
00:40:31,080 --> 00:40:33,730
axes should be labeled.

598
00:40:33,730 --> 00:40:36,570
I can't tell you the number of
times I've had a graduate

599
00:40:36,570 --> 00:40:40,890
student show up in my office,
having worked for weeks

600
00:40:40,890 --> 00:40:44,530
producing some data, put a plot
on my desk, and say, look

601
00:40:44,530 --> 00:40:46,570
at this, isn't it great.

602
00:40:46,570 --> 00:40:49,940
And I say, I have no
idea what it means.

603
00:40:49,940 --> 00:40:52,560
And sometimes, I'll say, well
what is the y-axis, and they

604
00:40:52,560 --> 00:40:53,680
end up scratching their head.

605
00:40:53,680 --> 00:40:56,700
They're not quite sure.

606
00:40:56,700 --> 00:40:58,390
You've got to label your axes.

607
00:40:58,390 --> 00:40:59,590
You've got to put a title.

608
00:40:59,590 --> 00:41:02,575
You've got to give the
person a break.

609
00:41:02,575 --> 00:41:05,000
Well how do we do that?

610
00:41:05,000 --> 00:41:07,750
Well it's pretty simple.

611
00:41:07,750 --> 00:41:09,000
So here's this code.

612
00:41:15,530 --> 00:41:23,770
So I just want to get rid of
the old graphs, because

613
00:41:23,770 --> 00:41:26,035
sometimes if you look at them,
it causes you problems.

614
00:41:30,720 --> 00:41:32,770
Because it's now hung.

615
00:41:32,770 --> 00:41:34,790
It won't continue until
I get rid of it.

616
00:41:34,790 --> 00:41:37,110
Now my shell is back.

617
00:41:37,110 --> 00:41:43,090
So pyLab.title just says, OK,
I'm going to call it 5%

618
00:41:43,090 --> 00:41:45,500
growth, compounded annually.

619
00:41:45,500 --> 00:41:49,570
Notice that I've put in that
it's 5% interest rate and that

620
00:41:49,570 --> 00:41:51,340
I'm compounding it
annually, not

621
00:41:51,340 --> 00:41:52,710
semi-annually or quarterly.

622
00:41:55,790 --> 00:41:59,280
The x-axis is going to be the
years of compounding and the

623
00:41:59,280 --> 00:42:02,760
y-axis, the value of the
principle in dollars.

624
00:42:11,450 --> 00:42:15,350
So now it's the same
curve, but a far

625
00:42:15,350 --> 00:42:17,455
more informative picture.

626
00:42:20,580 --> 00:42:23,730
All right here's where
I want to stop today.

627
00:42:23,730 --> 00:42:27,960
We're going to come back to this
topic, in probably more

628
00:42:27,960 --> 00:42:31,640
detail than you want, and spend
quite a lot of time

629
00:42:31,640 --> 00:42:35,890
talking about how do we produce
beautiful plots, and

630
00:42:35,890 --> 00:42:38,080
more importantly, how do we
produce plots that are,

631
00:42:38,080 --> 00:42:41,760
actually, meaningful to
those reading them.

632
00:42:41,760 --> 00:42:43,220
Thanks a lot.

633
00:42:43,220 --> 00:42:44,710
I'll see you in the
next lecture.