Answer to Question #330175 in Statistics and Probability for Robera

Question

Let A and B be two events associated with an experiment and suppose that P(A)=0.4 while P(AUB)=0.7. Let

P(B)=P

a) For what choice of P are A and B mutually exclusive?

b) For what choice of P are A and B independent?

Answer 1

a) For mutually exclusive events the following equality is true:

$P(A\cup B)=P(A)+P(B),\\ P=P(B)=P(A\cup B)-P(A)=0.7-0.4=0.3.$

b) For joint events

$P(A\cup B) =P(A)+P(B)-P(A\cap B).$

For independent events

$P(A\cap B)=P(A)\cdot P(B).$

So,

$P(A\cup B) =P(A)+P(B)-P(A)\cdot P(B),\\ 0.7=0.4+P-0.4P,\\ 0.6P=0.3,\\ P=0.5.$