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.