Video Transcript
Suppose that 𝐴 and 𝐵 are two
events. Given that the probability of 𝐴 is
equal to 0.52 and the probability of 𝐵 given 𝐴 is equal to 0.75, find the
probability of 𝐴 intersection 𝐵.
So, the first thing we want to do
is highlight what some of the notation means. So, first of all, if we think about
probability of 𝐴, if we have a look at this Venn diagram, well, then in the Venn
diagram, this circle here that I’ve shaded in would be the area that will be
representing the probability of 𝐴. And then if we take a look at 𝐴
intersection 𝐵, what this means would be the area that is the intersection between
the two circles, so 𝐴 and 𝐵. So, it’ll be this area here that
I’ve crosshatched in.
Well, then finally, if we look at
probability of 𝐵 given 𝐴, then what does this actually mean? Well, if we think about what this
means is the probability that 𝐵 happens given that 𝐴 has occurred. So, therefore, what we can do is
delete out the rest of our Venn diagram because nothing else matters now. We just want to look at the area
that is where 𝐴 occurs. And then within that, we want to
see what the probability that 𝐵 would occur is. But how do we work this out? So, how do we find the value?
Well, in fact, what we have is a
formula to help us. And that formula tells us that the
probability of 𝐵 given 𝐴 occurs is equal to the probability of 𝐴 intersection 𝐵
divided by the probability of 𝐴. But in this question, we’ve already
said we want to find what’s the probability of 𝐴 intersection 𝐵. So, therefore, to find this, all we
need to do is apply a bit of simple algebra. And when we rearrange our formula,
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, if we plug in our
values, what we’re gonna get is the probability of 𝐴 intersection 𝐵 is equal to
0.75 multiplied by 0.52. And what this gives us is the
answer to the question, what is the probability of 𝐴 intersection 𝐵? And it’s equal to 0.39.
