### Video Transcript

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

So first of all, I’m gonna jot down the key information from the question. So first of all, we know that the probability of 𝐴 is equal to 0.39. And we know that the probability of 𝐵 happening given that 𝐴 has happened is equal to 0.88. And when we look at the notation, it’s this vertical line that tells us that we’re dealing with conditional probability because it tells us that what we want is the probability of 𝐵 happening given that 𝐴 has happened.

And what the question wants us to find is the probability that 𝐵 doesn’t happen given that 𝐴 has happened. And this little bit of notation here, this dash that we call prime, means naught, so in this case naught 𝐵.

So if we consider the probability of 𝐵 happening given that 𝐴 happens or the probability of 𝐵 not happening given that 𝐴 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, it’s got to be equal to one. So therefore, as we know the probability of 𝐵 happening given that 𝐴 happens is equal to 0.88, we can say that 0.88 plus the probability of 𝐵 not happening given that 𝐴 happens is equal to one. So therefore, the probability of 𝐵 not happening given that 𝐴 happens is gonna be equal to one minus 0.88. And that’s 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 you’re doing it mentally is to count up. So let’s think of 88. We want to get to one. So we we’re 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, it’s gonna be 0.12 is the answer.

So therefore, we can say that the probability of 𝐵 not happening given that 𝐴 happens is equal to 0.12.