Video Transcript
Suppose the probability of 𝐵 given
𝐴 equals a half and the probability of 𝐴 equals three-sevenths. What is the probability that events
𝐴 and 𝐵 both occur?
Well, what this problem is, is a
conditional probability problem. And that is because we’ve got a
condition here. We’re looking at the probability
that 𝐵 occurs given that 𝐴 occurs. So therefore, what we’re gonna do
is use one of the formulae we have to actually solve conditional probability
problems. And the formula that we’re going to
use is this one. The probability of 𝐵 given 𝐴 is
equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐴.
Well, if we look at what we want to
find in this problem, we want to find out the probability that the events 𝐴 and 𝐵
both occur. But what this is, is the property
of 𝐴 and 𝐵. And this is the same as the
probability of 𝐴 intersection 𝐵. And this is what we’ve got
here. So all we need to do now is use
some algebra to rearrange our formula. And if we rearrange the formula to
make the probability of 𝐴 intersection 𝐵 our subject, then what we’re gonna get is
the probability of 𝐴 intersection 𝐵 is equal to the probability of 𝐵 given 𝐴
multiplied by the probability of 𝐴.
So therefore, the probability of 𝐴
intersection 𝐵 is going to be equal to a half multiplied by three-sevenths. And if we remind ourselves how we
multiply fractions, what we do is multiply the numerators and multiply the
denominators. Well, this is gonna be equal to
three fourteenths. So therefore, we can say that if we
suppose the probability of 𝐵 given 𝐴 is equal to a half and the probability of 𝐴
is equal three-sevenths, then the probability that events 𝐴 and 𝐵 both occur is
three fourteenths.
