Video: Calculating Conditional Probability to Decide Whether Events Are Independent

Video Transcript

Suppose the probability of event 𝐴 happening is equal two-fifths and the probability of event 𝐵 equals 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.

So in order to solve the first part of this problem, what we can do is use the conditional probability formula. So in order to use it, what we’re gonna do is substitute in the values we know. So first of all, we know that the probability of 𝐴 and 𝐵 is a fifth. And then this is gonna be divided by the probability of 𝐵, which is three-sevenths, which if we use our rules for fractions is the same as one-fifth multiplied by seven-thirds.

So in order to calculate what this is gonna be, we multiply the numerators and the denominators, which gives us seven-fifteenths. So therefore, we can say the probability of 𝐴 given 𝐵 is seven fifteenths.

Now, for the second part of the question, we’re asked to determine whether they’re independent. Well, if events 𝐴 and 𝐵 are independent, then the probability of 𝐴 given 𝐵 is gonna be equal to the probability of 𝐴. Well, the probability of 𝐴 is equal to two-fifths. Well, to compare this with the probability of 𝐴 given 𝐵, we’re gonna convert this into fifteenths. So to do that, we multiply the numerator and denominator by three. So that gives us six fifteenths. So therefore, as six fifteenths is not equal to seven fifteenths, events 𝐴 and 𝐵 are not independent.