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## Probability using combinatorics

# Exactly three heads in five flips

## Video transcript

So let's start again
with a fair coin. And this time, instead of
flipping it four times, let's flip it five times. So five flips of this fair coin. And what I want to think
about in this video is the probability of
getting exactly three heads. And the way I'm
going to think about it is, if you have five flips,
how many different equally likely possibilities are there? So you're going to have the
first flip-- let me draw it over here-- first flip. And there's two
possibilities there. It could be heads or tails. Second flip, two
possibilities there. Third flip, two possibilities. Fourth flip, two possibilities. Fifth flip, two possibilities. So it's 2 times 2 times
2 times 2 times 2. I hope I said that five times. Or you could view that as
2 to the fifth power. And that is going
to be equal to 32 equally likely possibilities. 32-- 2 times 2 is 4, 4 times
2 is 8, 8 times 2 is 16, 16 times 2 is 32 possibilities. And to figure out
this probability, we really just have to
figure out how many of those possibilities involve
getting three heads. We could write out all
the 32 possibilities and literally just
count the heads. But let's just use
that other technique that we just started to
explore in that last video. We have five flips here. So let me draw the flips--
one, two, three, four, five. And we want to have
exactly three heads. And I'm going to call
those three heads-- let me do it in pink-- heads
A, heads B, heads C, just to give them a name. Although what we're going
to see later in this video is that we don't want to
differentiate between them. To us, it makes no difference if
we get this ordering-- heads A, heads B, heads C, tails, tails. Or if we get this
ordering-- heads A, heads C, or heads
B, tails, tails. We can't count these as
two different orderings. We can only count this as one. So what we're going
to do is first come up with all of
the different orderings if we cared about the
difference between A, B, and C. And then we're going to divide
by all of the different ways that you can arrange
three different things. So how many ways
can we put A, B, and C into these five buckets
that we can view as the flips, if we cared about A, B, and C ? So let's start with A.
If we haven't allocated any of these buckets to
any of the heads yet, then we could say that A could
be in five different buckets. So there's five possibilities
where A could be. So let's just say that this
is the one that it goes in, although it could be in
any one of these five. But if this takes
up one of the five, then how many different
possibilities can this heads sit in? How many different
possibilities are there? Well, then there's only going
to be four buckets left. So then there's only
four possibilities. And so if this was
where heads A goes, then heads B could be in
any of the other four. If heads A was in
this first one, then heads B could have been
in any of the four. I'll just do a
particular example. Maybe heads B shows
up right there. So once we've taken
two of the slots, how many spaces do
we have for heads C? Well, we only have three
spaces left, then, for heads C. And so it could be in any
of these three spaces. And just to show a
particular example, it would look like that. And so if you cared about order,
how many different ways can you, out of five
different spaces, allocate them to
three different heads? You would say it is
5 times 4 times 3. 5 times 4 is 20, times
3 is equal to 60. So you would say there
are 60 different ways to arrange heads A, B, and C
in five buckets, or five flips, or if these were
people, in five chairs. And obviously, there
aren't 60 possibilities of getting three heads. In fact, there's only 32
equally likely possibilities. And the reason why we got
such a big number over here is that we are counting this
scenario as being fundamentally different than if this was
heads B, heads A, and then heads C over here. And what we need to
do is say, well, these aren't different possibilities. We don't have to
overcount for all of the different ways
you arrange this. And so what we need
to do is divide this by all of the
different ways that you can arrange three things. So if I have three things
that are in three spaces. So here I have a heads in
the second flip, third flip, and fifth flip. If I have three things in
three spaces like this, how many ways can
I arrange them? And so if I have three
spaces, how many ways can I arrange an A, B, and C
in those three spaces? Well, A can go
into three spaces. It can go into any of the three. Then B would have two spaces
left, once A takes one of them. And then C would
have one space left, once A and B take two of them. So there's 3 times 2
times 1 way to arrange three different things. So 3 times 2 times
1 is equal to 6. So the number of possibilities
of getting three heads is actually going to be
this 5 times 4 times 3. Let me write this
down in another color. So the number of possibilities--
let's write "poss" for short-- is equal to this 5 times 4
times 3 over the number of ways that I can rearrange
three things. Because we don't want to
overcount for all of these, viewing this arrangement
as fundamentally different than this arrangement. So then we want to
divide it by-- I want to do that same orange
color-- dividing it by 3 times 2 times 1. And which gives us, in the
numerator, 120 divided by 6. Oh sorry, that's not--
it's 60 divided by 6. This is 60. 5 times 4 times 3 is 60. It gives us 60 divided
by 6, which gives us 10 possibilities that gives
us exactly three heads. And that's of 32 equally
likely possibilities. So the probability of getting
exactly three heads-- well, you get exactly three
heads in 10 of the 32 equally likely possibilities. So you have a 5/16
chance of that happening.