### Video Transcript

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

Well, first, let us recall the
conditional probability formula. The probability of π΄ given π΅ is
equal to the probability of π΄ intersection π΅ over the probability of π΅. Well, the probability of π΄
intersection π΅ is the probability that event π΄ occurs and event π΅ 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 π΅ occurring is three-sevenths. So the probability of π΄ given π΅
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, thereβs the answer to our
first part of the question. The probability of π΄ given π΅ is
equal to seven-fifteenths.

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

Now, if two-fifths is the same as
seven-fifteenths, then weβd need to go on to check whether the probability of π΅ is
equal to the probability of π΅ given π΄. 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
π΄ is not equal to the probability of π΄ given π΅. And the fact that we need both of
these conditions to be true to prove the independence of events π΄ and π΅, the fact
that weβve shown that the probability of π΄ is not equal to the probability of π΄
given π΅, means that we know that these two events arenβt independent. We donβt even need to go on to
check whether the probability of π΅ is equal to the probability of π΅ given π΄.