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ALBERT MEYER: Today's
topic is random variables.

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Random variables are an
absolutely fundamental concept

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in probability theory.

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But before we get into
officially defining them,

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let's start off
with an example that

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in fact, is a game because
that's a fun way to start.

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So we're going to play
the bigger number game

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and here's how it works.

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There are two teams,
and Team 1 has

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the task of picking two
different integers between 0

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and 7 inclusive, and
they write one integer

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on one piece of paper
and the other integer

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on the other piece of paper.

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They turn the two
pieces of paper

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face down so the
numbers are not visible,

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and the other team then sees
these two pieces of paper

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whose other side has
different numbers written

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on them sitting on the table.

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What Team 2 then does is picks
one of the pieces of paper

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and turns it over and
looks at the number on it.

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And then, based on
what that number

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is, they make a decision,
stick with the number

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they have or switch to the
other unknown number on the face

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down piece of paper.

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And that'll be
their final number.

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And the game is that Team
2 wins if they wind up

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with the larger number.

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So they're going to look
at the number on the paper

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that they expose
and they're going

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to try to decide
whether it looks

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like a big number
or little number.

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If it looks like a big
number, they'll stick with it.

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If it looks like
a little number,

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they'll switch to the other
one that they hope is larger.

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So which team do you think
has an advantage here?

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Course, if you've read
the notes, you know.

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But if you haven't been
exposed to this before,

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it's not really so obvious.

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And what we
encourage and what we

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used to do when we ran this
in real time in classes

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that we would have
students in teams, split

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their team in half,
one would be Team 1

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and the other would
be Team 2, and they'd

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play the game few times,
see if they could figure out

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who had the advantage.

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And if you have the
opportunity, this

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might be a good moment
to stop this video

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and try playing the game with
some friends if they're around.

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Otherwise, let's just proceed
and see how it all works.

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So this is the strategy
Team 2 is going to adopt.

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They're going to take this
idea about big and small

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that I mentioned and act
on it in a methodical way.

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So they're going to pick
a paper to expose, giving

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each paper equal probability.

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So that guarantees
that they have

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a 50/50 chance of picking the
big number and a 50/50 chance

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of picking the little number.

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Whatever ingenuity Team 1 tried
to do on which piece of paper

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was on the left and
which was on the right,

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it doesn't really matter if Team
2 simply picks a piece of paper

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at random.

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There's no way
that Team 1 can try

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to fake out Team 2 on
where they put the number.

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OK.

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The next step is
that Team 2 is going

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to decide whether the
number that they can see,

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the exposed number, is small.

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And if so, would they switch?

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And otherwise they stick.

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So that is, they're going to
define some threshold Z where

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being less than or
equal to Z means small,

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and being greater
than Z means large.

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The question is, how
do they choose Z?

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Well, a naive thing
to do would be

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to choose Z to be in the middle
of the interval from 0 to 7.

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Let's say, you
choose Z equals 3.

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So there would be four numbers
less than or equal to Z

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and four numbers greater
than Z. But of course,

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as soon as Team 1 knew that that
was your Z, what would they do?

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Well, they would make sure
that both numbers were

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on the same side of
Z. If you're Z was 3,

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they would always choose their
numbers to be, say, 0 and 1.

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And that way, when
you were switching,

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your Z would tell you that
you had a small number,

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you should switch
to the other one.

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And you'd only have a
50/50 chance of winning.

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So if you fixed that
value of Z, Team 2

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has a way of ensuring that
you have no advantage.

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You can only win with
probability 50/50.

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And that's true no
matter what Z you take.

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If Team 1 knew what
your Z was, they

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would just make sure
to pick their two

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numbers on the same
side of your Z.

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And then your Z wouldn't
really tell you anything.

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You'd switch or
stick in both cases,

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and you'd only
have a 50/50 chance

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of picking the right number.

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So what you do-- and this is
where probability comes in--

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is you pick Z in a way that
can't be predicted or made

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use of by Team 1.

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You pick Z at random, to
be any number from 0 to 7,

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not including 7 including 0.

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That is, your number is
either 0, 1, 2, up through 6.

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And being less than or
equal to Z means small,

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and being greater
than Z means large.

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And when you see a small
number, you'll switch

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and when you see a large
number, you'll stick.

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But what's going to be large
and what's going to be small

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is going to vary each time
you play the game, depending

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on what random number,
Z, comes out to be.

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So let's analyze your
probability if you're Team 2.

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What's the probability that
you're going to win now?

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Well, let's suppose that Team
1 picks these two numbers.

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We don't know what
they are, but they

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have to pick a low number
that's less than a high number.

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So these two numbers
are at least 1 apart,

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they can't have the same
number on both pieces of paper.

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Otherwise, it's clear
that you are not

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going to be able to
pick the large one, that

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would be cheating.

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OK, so there's two
different numbers.

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So one of them has to
be less than the other.

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We don't know how much
less, might be a lot less,

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might be only one less,
but low is less than high.

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OK, now we can consider
three cases of what

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happens with your strategy.

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The most interesting
case is the middle case.

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That is, when your Z,
which was chosen at random,

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happens to fall in the
interval between low and high.

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That is, your Z is strictly
less than high and greater than

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or equal to low.

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And then in that case, your Z
is really guiding you correctly

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on what to do.

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If you turn over the
low card, then it's

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going to look low because
it's less than or equal to Z

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so you'll switch to
the high card and win.

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If you turn over
the high card, it's

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going to be greater than
Z so it'll look high

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and you'll know
to stick with it.

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So in this case, you're
guaranteed to win.

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If you were lucky
enough to guess

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the right threshold between low
and high, you're going to win.

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And so the probability
that you win,

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given the middle
case occurs, is 1.

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Now, what about the middle case?

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How often does that happen?

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Well, the difference between
low and high is at least 1,

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so there's guaranteed
to be 1 chance in 7

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that your Z is going
to fall between them.

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And it could be more if low
and high are further apart,

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but as long as they're
at least one apart,

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there's a 1/7 chance that you're
going to fall in between them.

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OK.

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Now, in case H,
that's the case where

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Z happens to be
chosen greater than

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or equal to the high
number that Team 1 shows.

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In other words, Z is bigger
than both numbers than Team 1

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shows and put on
the pieces of paper.

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Well, in that case, Z just
isn't telling you anything.

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So what's going to happen is
that both numbers are going

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to look high to you--
sorry-- both numbers

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are going to look low to you
because they're both less than

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or equal to Z. So you'll switch.

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And that means that
you'll win, if and only

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if, you happen to turn
the low card over first.

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Well that was 50/50.

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So the probability
that you win, given

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that Z-- both cards are
on the low side of Z,

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you'll win with half the time.

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And symmetrically, if Z is
less than the low card, that

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is, Z is less than both
cards chosen by Team 1,

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then they're both going to
look high, and so you'll stick.

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And that means that you'll
stick, you'll win, if and only

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if, you happen to have
picked the high card.

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There's a 50/50 chance of that.

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So again, in this case that
Z makes both cards look high,

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or Z itself is low, Team 2,
you win with probability 1/2.

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Well, that's great because now
we can apply total probability.

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And what total probability
tells us is that Team 2 wins

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is the probability that
they win given case

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M times the probability
of M plus the probability

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that they win given
not the middle case

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times the probability
of not the middle case.

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But we figured out
what these were.

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Well, at least
inequalities on them,

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because there's
probability 1 that you'll

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win 1/7 of the time.

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And there's probability
a 1/2 that you'll

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win the rest of the time,
the other 6/7 of the time.

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You're going to win
4/7 of the time.

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The probability that you win
playing your strategy is 4/7.

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It's better than 50/50.

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You have an advantage.

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And whether that was a priori
obvious or not, I don't know.

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But I think it's kind of cool.

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OK, you win with
probability 4/7.

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Now, Team 2 has the advantage.

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And the important
thing to understand

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is it does not matter
what team does.

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No matter how smart
Team 1 is, Team 2

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has gotten control
of the situation

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because they picked--
which piece of paper

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they picked at random 50/50.

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So it doesn't matter
what strategy Team 1 used

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on where they
placed the numbers.

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And they chose Z
randomly, so again, it

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doesn't matter what
numbers Team 1 shows.

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Team 2 is still going to have
their 1/7 chance of coming out

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ahead, which is enough to tip
the balance in their favor.

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It's interesting that
symmetrically, Team 1 also

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has a random strategy
that they can use,

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which guarantees that no
matter what Team 2 does, Team 2

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wins with probability
at most 4/7.

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So either team can
force the probability

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that Team 2 wins to be at
most 4/7 and at least 4/7.

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So if they both play optimally,
it's going to stay at 4/7.

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And that's again, true no
matter what Team 2 does,

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Team 1 can put this
upper bound to 4/7 on it.

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So essentially we can
say that the value

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of this game, the
probability that Team 2 wins

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is optimally for both is 4/7.

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OK, now what does this game
got to do with anything,

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with our general topic
of random variables?

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Well, we'll be
formal in a moment.

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But informally,
a random variable

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is simply a number that's
produced by a random process.

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And just to give an
example before we come up

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with a formal definition,
the threshold variable Z

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was a thing that took a
value from 0 to 6 inclusive,

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each with probability 1/7.

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So it was producing a
number by a random process,

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that chose a number at random
with equal probability.

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If Team 2 plays properly
at random picking which

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piece of paper to expose, then
the number of the exposed card,

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or more precisely, whether the
exposed card is high or low,

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will also be a random variable.

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And if Team 1 plays optimally,
the number on the exposed card

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is going to be a
random variable.

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That is, Team 1 in their
optimal strategy that

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00:11:42,350 --> 00:11:46,310
puts an upper bound to 4/7 is
in fact, going to choose the two

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00:11:46,310 --> 00:11:47,230
numbers randomly.

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00:11:47,230 --> 00:11:49,040
So the exposed card
is going to wind up

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00:11:49,040 --> 00:11:51,930
being another random
variable, a number produced

244
00:11:51,930 --> 00:11:53,270
by the random process.

245
00:11:53,270 --> 00:11:55,800
And likewise, the number
of the larger card

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00:11:55,800 --> 00:12:00,070
if Team 1 picks its larger
and smaller cards randomly,

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00:12:00,070 --> 00:12:02,560
it's going to be another
example of a number produced

248
00:12:02,560 --> 00:12:04,256
by a random process.

249
00:12:04,256 --> 00:12:06,130
And likewise, the number
of the smaller card.

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00:12:06,130 --> 00:12:07,230
So that's enough examples.

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00:12:07,230 --> 00:12:09,030
This little game
has a whole bunch

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00:12:09,030 --> 00:12:10,940
of random variables
appearing in it.

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00:12:10,940 --> 00:12:13,480
And in the next
segment, we will look

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00:12:13,480 --> 00:12:15,810
again officially,
what is the definition

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00:12:15,810 --> 00:12:18,030
of a random variable?