### Video Transcript

Suppose that ๐ด and ๐ต are events in a random experiment. Given that the probability of ๐ด is equal to 0.39 and the probability of ๐ต given ๐ด is equal to 0.88, find the probability of naught ๐ต given ๐ด.

So first of all, Iโm gonna jot down the key information from the question. So first of all, we know that the probability of ๐ด is equal to 0.39. And we know that the probability of ๐ต happening given that ๐ด has happened is equal to 0.88. And when we look at the notation, itโs this vertical line that tells us that weโre dealing with conditional probability because it tells us that what we want is the probability of ๐ต happening given that ๐ด has happened.

And what the question wants us to find is the probability that ๐ต doesnโt happen given that ๐ด has happened. And this little bit of notation here, this dash that we call prime, means naught, so in this case naught ๐ต.

So if we consider the probability of ๐ต happening given that ๐ด happens or the probability of ๐ต not happening given that ๐ด happens, well these are in fact exhaustive because this covers every eventuality. So therefore, what we can say is that if we add their probabilities together, itโs got to be equal to one. So therefore, as we know the probability of ๐ต happening given that ๐ด happens is equal to 0.88, we can say that 0.88 plus the probability of ๐ต not happening given that ๐ด happens is equal to one. So therefore, the probability of ๐ต not happening given that ๐ด happens is gonna be equal to one minus 0.88. And thatโs because we subtracted 0.88 from each side. Well, this is gonna be equal to 0.12.

And the best way to work this out if youโre doing it mentally is to count up. So letโs think of 88. We want to get to one. So we weโre going to get to 100. We add twos to get to 90 and then add another 10 to get to 100. So we add 12. So therefore, itโs gonna be 0.12 is the answer.

So therefore, we can say that the probability of ๐ต not happening given that ๐ด happens is equal to 0.12.