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## Multiplication rule for independent events

# Compound probability of independent events

AP.STATS:

VAR‑4 (EU)

, VAR‑4.E.1 (EK)

CCSS.Math: , ## Video transcript

Let's think about
the situation where we have a completely
fair coin here. So let me draw it. I'll assume it's a
quarter or something. Let's see. So this is a quarter. Let me draw my best attempt at
a profile of George Washington. Well, that'll do. So it's a fair coin. And we're going to flip
it a bunch of times and figure out the
different probabilities. So let's start with a
straightforward one. Let's just flip it once. So with one flip
of the coin, what's the probability
of getting heads? Well, there's two equally
likely possibilities. And the one with heads is one
of those two equally likely possibilities, so
there's a 1/2 chance. Same thing if we
were to ask what is the probability
of getting tails? There are two equally likely
possibilities, and one of those gives us tails, so 1/2. And this is one
thing to realize. If you take the
probabilities of heads plus the probabilities
of tails, you get 1/2 plus 1/2, which is 1. And this is generally true. The sum of the probabilities
of all of the possible events should be equal to 1. And that makes sense,
because you're adding up all of these fractions,
and the numerator will then add up to all
of the possible events. The denominator is always
all the possible events. So you have all
the possible events over all the
possible events when you add all of these things up. Now let's take it up a notch. Let's figure out the
probability of-- I'm going to take this
coin, and I'm going to flip it twice--
the probability of getting heads and then
getting another heads. There's two ways
to think about it. One way is to just
think about all of the different possibilities. I could get a head on
the first flip and a head on the second flip,
head on the first flip, tail on the second flip. I could get tails on the first
flip, heads on the second flip. Or I could get
tails on both flips. So there's four distinct,
equally likely possibilities. And one way to think about
is on the first flip, I have two possibilities. On the second flip, I have
another two possibilities. I could have heads or
tails, heads or tails. And so I have four
possibilities. For each of these possibilities,
for each of these two, I have two possibilities here. So either way, I have four
equally likely possibilities. And how many of those
meet our constraints? Well, we have it
right over here, this one right
over here-- having two heads meets our constraints. And there's only one
of those possibilities. I've only circled one
of the four scenarios. So there's a 1/4 chance
of that happening. Another way you could
think about this-- and this is because these are
independent events. And this is a very
important idea to understand in
probability, and we'll also study scenarios that
are not independent. But these are
independent events. What happens in the
first flip in no way affects what happens
in the second flip. And this is actually one thing
that many people don't realize. There's something called
the gambler's fallacy, where someone thinks if I
got a bunch of heads in a row, then all
of a sudden, it becomes more likely on the
next flip to get a tails. That is not the case. Every flip is an
independent event. What happened in the
past in these flips does not affect the
probabilities going forward. The fact you got a heads
on the first flip in no way affects that you got a
heads on the second flip. So if you can make
that assumption, you could say that the
probability of getting heads and heads, or
heads and then heads, is going to be the same thing
as the probability of getting heads on the first flip
times the probability of getting heads
on the second flip. And we know the probability of
getting heads on the first flip is 1/2 and the probability
of getting heads on the second flip is 1/2. And so we have 1/2
times 1/2, which is equal to 1/4, which is
exactly what we got when we tried out all of the
different scenarios, all of the equally
likely possibilities. Let's take it up another notch. Let's figure out
the probability-- and we've kind of
been ignoring tails, so let's pay some
attention to tails. The probability of
getting tails and then heads and then tails-- so
this exact series of events. So I'm not saying in any
order two tails and a head. I'm saying this exact order--
the first flip is a tails, second flip is a heads, and
then third flip is a tail. So once again, these are
all independent events. The fact that I got tails
on the first flip in no way affects the
probability of getting a heads on the second flip. And that in no way
affects the probability of getting a tails
on the third flip. So because these are
independent events, we could say that's the same
thing as the probability of getting tails on
the first flip times the probability of getting
heads on the second flip times the probability of getting
tails on the third flip. And we know these are
all independent events, so this right over here is
1/2 times 1/2 times 1/2. 1/2 times 1/2 is 1/4. 1/4 times 1/2 is equal to
1/8, so this is equal to 1/8. And we can verify it. Let's try out all of the
different scenarios again. So you could get
heads, heads, heads. You could get
heads, heads, tails. You could get
heads, tails, heads. You could get
heads, tails, tails. You can get tails, heads, heads. This is a little
tricky sometimes. You want to make sure
you're being exhaustive in all of the different
possibilities here. You could get
tails, heads, tails. You could get
tails, tails, heads. Or you could get
tails, tails, tails. And what we see here is
that we got exactly eight equally likely possibilities. We have eight equally
likely possibilities. And the tail, heads, tails
is exactly one of them. It is this possibility
right over here. So it is 1 of 8 equally
likely possibilities.