Question Video: Calculating Conditional Probability to Decide Whether Events Are Independent Mathematics

Video Transcript

Suppose the probability of 𝐴 is two-fifths and the probability of 𝐵 is three-sevenths. The probability that event 𝐴 occurs and event 𝐵 also occurs is one-fifth. Calculate the probability of 𝐴 given 𝐵 and then evaluate whether events 𝐴 and 𝐵 are independent.

Well, first, let us recall the conditional probability formula. The probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 over the probability of 𝐵. Well, the probability of 𝐴 intersection 𝐵 is the probability that event 𝐴 occurs and event 𝐵 also occurs, which we were given in the question is a fifth. And we were also told in the question that the probability of event 𝐵 occurring is three-sevenths. So the probability of 𝐴 given 𝐵 is a fifth divided by three-sevenths. And, of course, an equivalent operation to dividing by a fraction is multiplying by the reciprocal of that fraction. So this is equal to a fifth times seven over three. Well, there’s the answer to our first part of the question. The probability of 𝐴 given 𝐵 is equal to seven-fifteenths.

For the second part of the question, let’s recall our test for independence using the conditional probability formulae. If the probability of 𝐴 is equal to the probability of 𝐴 given 𝐵 and the probability of 𝐵 is equal to the probability of 𝐵 given 𝐴, then we can say that events 𝐴 and 𝐵 are independent. Now the question told us that the probability of 𝐴 is two-fifths. So the next step is to work out the probability of 𝐴 given 𝐵. Well, in fact, that’s what we worked out in the first part of the question. So we know that the probability of 𝐴 given 𝐵 is seven-fifteenths.

Now, if two-fifths is the same as seven-fifteenths, then we’d need to go on to check whether the probability of 𝐵 is equal to the probability of 𝐵 given 𝐴. Now to compare two-fifths and seven-fifteenths, we need to get a common denominator. We can do this by multiplying the numerator and denominator by three and two-fifths becomes six-fifteenths. That means that the probability of 𝐴 is not equal to the probability of 𝐴 given 𝐵. And the fact that we need both of these conditions to be true to prove the independence of events 𝐴 and 𝐵, the fact that we’ve shown that the probability of 𝐴 is not equal to the probability of 𝐴 given 𝐵, means that we know that these two events aren’t independent. We don’t even need to go on to check whether the probability of 𝐵 is equal to the probability of 𝐵 given 𝐴.