Answer to Question #166478 in Statistics and Probability for phemelo mongae

Question

If A and B are events such that P(A) = 0.6, P(B|A) = 0.3 and P(A ∪ B) = 0.72. Are A and B independent, mutually exclusive or both?

Answer 1

Solution:

We are given $P(A), P(B|A), P(A+B)$

If $P(B|A)=P(B)$ We can say that events are independen.Therefore, we need to find $P(B)$. To do so we need to remember the formula of sum of events:

$P(A+B) = P(A) + P(B) - P(A*B)$

$P(B) = P(A+B) + P(A*B) - P(A)$

And $P(A*B) = P(A)*P(B|A)= 0.6 * 0.3 = 0.18$, so:

$P(B) = 0.72 + 0.18 - 0.6=0.3$

$P(B) = 0.3 = P(B|A),$ therefore the events are independent.

For events to be mutually exclusive, they can not occur at the same time, so following equation has to occur:

$P(A*B) = 0$,

but we previously found: $P(A*B) = 0.18$, therefore the events are not mutually exclusive.

Answer:

Events are independent, but not mutually exclusive