Question

suppose events A and B are such that p(A)=0.25,P(B) =0.3 and P(B|A)= 0.5
i) compute P(AnB) and P(AuB)
ii) Are events A and B independent?
iii) Are events A and B mutually exclusive?
b)if P(A)=0.6, P(B)= 0.15 and P(B|A')=0.25, find the following probabilities
I)P(B|A), P(A|B),P(AuB)

Answer 1

A) I) P(A B) = P(B|A) * P(A)

= 0.5 * 0.25 = 0.125

P(A U B) = P(A) + P(B) - P(A B)

= 0.25 + 0.3 - 0.125 = 0.425

ii) P(A) * P(B) = 0.25 * 0.3 = 0.075

Since P(A) * P(B) P(A B) , so the events A and B are not independent.

iii) since P(A B) 0, so the events A and B are not mutually exclusive.

B) i) P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Or, 0.15 = P(B|A) * 0.6 + 0.25 * 0.4

Or, 0.15 = P(B | A) * 0.6 + 0.1

Or, P(B | A) = (0.15 - 0.1)/0.6

Or, P(B | A) = 0.083

P(A|B) = P(B|A) * P(A)/P(B)

= (0.083 * 0.6)/0.15 = 0.332

P(A U B) = P(A) + P(B) - P(A B)

= 0.6 + 0.15 - (0.332 * 0.15)

= 0.6 + 0.15 - 0.0498 = 0.7002