Question Video: Finding Conditional Probabilities | Nagwa Question Video: Finding Conditional Probabilities | Nagwa

Question Video: Finding Conditional Probabilities

Suppose that ๐ด and ๐ต are events in a random experiment. Given that P(๐ด) = 0.39 and P(๐ต | ๐ด) = 0.88, find P(๐ตโ€ฒ | ๐ด).

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Video Transcript

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

So first of all, Iโ€™m gonna jot down the key information from the question. So first of all, we know that the probability of ๐ด is equal to 0.39. And we know that the probability of ๐ต happening given that ๐ด has happened is equal to 0.88. And when we look at the notation, itโ€™s this vertical line that tells us that weโ€™re dealing with conditional probability because it tells us that what we want is the probability of ๐ต happening given that ๐ด has happened.

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

So if we consider the probability of ๐ต happening given that ๐ด happens or the probability of ๐ต not happening given that ๐ด 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, itโ€™s got to be equal to one. So therefore, as we know the probability of ๐ต happening given that ๐ด happens is equal to 0.88, we can say that 0.88 plus the probability of ๐ต not happening given that ๐ด happens is equal to one. So therefore, the probability of ๐ต not happening given that ๐ด happens is gonna be equal to one minus 0.88. And thatโ€™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 youโ€™re doing it mentally is to count up. So letโ€™s think of 88. We want to get to one. So we weโ€™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, itโ€™s gonna be 0.12 is the answer.

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

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