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Let's do a little bit of
probability with playing cards.

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And for the sake of
this video, we're

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going to assume that our
deck has no jokers in it.

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You could do the same
problems with the joker,

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you'll just get slightly
different numbers.

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So with that out of
the way, let's first

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just think about
how many cards we

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have in a standard playing deck.

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So you have four
suits, and the suits

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are the spades, the diamonds,
the clubs, and the hearts.

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You have four suits and
then in each of those suits

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you have 13 different
types of cards--

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and sometimes it's
called the rank.

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You have the ace, then you have
the two, the three, the four,

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the five, the six,
seven, eight, nine, ten,

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and then you have the Jack,
the King, and the Queen.

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And that is 13 cards.

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So for each suit
you can have any

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of these-- you can
have any of the suits.

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So you could have a Jack of
diamonds, a Jack of clubs,

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a Jack of spades,
or a Jack of hearts.

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So if you just multiply
these two things--

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you could take a deck of playing
cards, take out the jokers

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and count them-- but
if you just multiply

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this you have four suits, each
of those suits have 13 types.

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So you're going to
have 4 times 13 cards,

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or you're going to have 52 cards
in a standard playing deck.

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Another way you could
have said, look,

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there's 13 of these
ranks, or types,

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and each of those come in four
different suits-- 13 times 4.

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Once again, you would
have gotten 52 cards.

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Now, with that of
the way, let's think

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about the probabilities
of different events.

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So let's say I
shuffle that deck.

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I shuffle it really,
really well and then

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I randomly pick a
card from that deck.

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And I want to think about
what is the probability that I

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pick a Jack.

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Well, how many equally
likely events are there?

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Well, I could pick any
one of those 52 cards.

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So there's 52 possibilities
for when I pick that card.

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And how many of those 52
possibilities are Jacks?

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Well you have the Jack of
spades, the Jack of diamonds,

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the Jack of clubs, and
the Jack of hearts.

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There's four Jacks in that deck.

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So it is 4 over 52-- these
are both divisible by 4-- 4

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divided by 4 is 1, 52
divided by 4 is 13.

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Now, let's think
about the probability.

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So I'll start over.

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I'm going to put that
Jack back and I'm

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going to reshuffle the deck.

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So once again, I
still have 52 cards.

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So what's the probability
that I get a hearts?

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What's the probability
that I just randomly pick

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a card from a shuffled
deck and it is a heart?

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Well, once again,
there's 52 possible cards

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I could pick from.

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52 possible, equally likely
events that we're dealing with.

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And how many of those
have our hearts?

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Well, essentially 13
of them are hearts.

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For each of those suits
you have 13 types.

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So there are 13
hearts in that deck.

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There are 13 diamonds
in that deck.

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There are 13 spades
in that deck.

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There are 13 clubs in that deck.

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So 13 of the 52 would result
in hearts, and both of these

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are divisible by 13.

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This is the same thing as 1/4.

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One in four times
I will pick it out,

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or I have a one in four
probability of getting a hearts

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when I randomly pick a card
from that shuffled deck.

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Now, let's do something that's
a little bit more interesting,

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or maybe it's a little obvious.

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What's the probability
that I pick something

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that is a Jack-- I'll just
write J-- and it is a hearts?

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Well, if you are reasonably
familiar with cards

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you'll know that
there's actually

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only one card that is
both a Jack and a heart.

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It is literally
the Jack of hearts.

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So we're saying, what
is the probability

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that we pick the exact
card, the Jack of hearts?

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Well, there's only
one event, one card,

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that meets this criteria
right over here,

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and there's 52 possible cards.

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So there's a one
in 52 chance that I

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pick the Jack of hearts--
something that is both a Jack

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and it's a heart.

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Now, let's do something a
little bit more interesting.

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What is the
probability-- you might

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want to pause this and think
about this a little bit

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before I give you the answer.

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What is the probability
of-- so I once again, I

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have a deck of 52
cards, I shuffled it,

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randomly pick a card from that
deck-- what is the probability

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that that card that I pick from
that deck is a Jack or a heart?

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So it could be the
Jack of hearts,

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or it could be the
Jack of diamonds,

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or it could be the
Jack of spades,

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or it could be the
Queen of hearts,

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or it could be
the two of hearts.

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So what is the
probability of this?

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And this is a little bit
more of an interesting thing,

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because we know, first
of all, that there

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are 52 possibilities.

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But how many of
those possibilities

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meet these conditions that
it is a Jack or a heart.

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And to understand that,
I'll draw a Venn diagram.

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Sounds kind of fancy,
but nothing fancy here.

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So imagine that this
rectangle I'm drawing here

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represents all of the outcomes.

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So if you want, you could
imagine it has an area of 52.

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So this is 52 possible outcomes.

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Now, how many of those
outcomes result in a Jack?

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So we already learned, one out
of 13 of those outcomes result

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in a Jack.

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So I could draw a
little circle here,

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where that area-- and I'm
approximating-- represents

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the probability of a Jack.

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So it should be
roughly 1/13, or 4/52,

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of this area right over here.

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So I'll just draw it like this.

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So this right over here is
the probability of a Jack.

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There's four possible
cards out of the 52.

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So that is 4/52,
or one out of 13.

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Now, what's the probability
of getting a hearts?

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Well, I'll draw another
little circle here

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that represents that.

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13 out of 52 cards
represent a heart.

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And actually, one of those
represents both a heart

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and a Jack.

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So I'm actually going
to overlap them,

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and hopefully this will
make sense in a second.

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So there's actually 13
cards that are a heart.

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So this is the number of hearts.

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And actually, let me write this
top thing that way as well.

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It makes it a little bit
clearer that we're actually

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looking at the number of Jacks.

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And of course,
this overlap right

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here is the number of Jacks
and hearts-- the number

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of items out of this 52 that
are both a Jack and a heart--

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it is in both sets here.

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It is in this green circle and
it is in this orange circle.

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So this right over here--
let me do that in yellow

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since I did that problem in
yellow-- this right over here

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is a number of Jacks and hearts.

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So let me draw a
little arrow there.

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It's getting a little
cluttered, maybe

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I should draw a little
bit bigger number.

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And that's an
overlap over there.

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So what is the probability
of getting a Jack or a heart?

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So if you think about
it, the probability

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is going to be the
number of events

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that meet these conditions,
over the total number events.

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We already know the total
number of events are 52.

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But how many meet
these conditions?

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So it's going to be the
number-- you could say,

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well, look at the green
circle right there says

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the number that gives us a Jack,
and the orange circle tells us

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the number that
gives us a heart.

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So you might want to say,
well, why don't we add up

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the green and the
orange, but if you

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did that, you would
be double counting,

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Because if you add
it up-- if you just

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did four plus 13--
what are we saying?

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We're saying that
there are four Jacks

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and we're saying that
there are 13 hearts.

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But in both of these, when we
do it this way, in both cases

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we are counting
the Jack of hearts.

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We're putting the
Jack of hearts here

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and we're putting the
Jack of hearts here.

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So we're counting
the Jack of hearts

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twice, even though there's
only one card there.

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So you would have to subtract
out where they're common.

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You would have to
subtract out the item that

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is both a Jack and a heart.

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So you would subtract out a 1.

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Another way to
think about it is,

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you really want to figure
out the total area here.

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And let me zoom in-- and I'll
generalize it a little bit.

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So if you have one
circle like that,

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and then you have another
overlapping circle like that,

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and you wanted to figure
out the total area of both

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of these circles
combined, you would

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look at the area of this circle.

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And then you could add it
to the area of this circle.

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But when you do that, you'll
see that when you add the two

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areas, you're counting
this area twice.

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So in order to only
count that area once,

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you have to subtract
that area from the sum.

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So if this area has
A, this area is B,

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and the intersection
where they overlap is C,

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the combined area is
going to be A plus B-- --

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minus where they
overlap-- minus C.

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So that's the same
thing over here,

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we're counting all
the Jacks, and that

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includes the Jack of hearts.

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We're counting all
the hearts, and that

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includes the Jack of hearts.

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So we counted the
Jack of hearts twice,

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so we have to subtract
1 out of that.

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This is going to be
4 plus 13 minus 1,

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or this is going to be 16/52.

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And both of these things
are divisible by 4.

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So this is going to be the
same thing as, divide 16 by 4,

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you get 4.

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52 divided by 4 is 13.

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So there's a 4/13 chance that
you'd get a Jack or a hearts.