Question

On the probability field {Ω, K, P}, consider two independent events A and B with P(A) = 1/2, P(B) = 2/5. Calculate the conditional probability that both A and B occur, given that at least one of the events A and B has occurred.

Answer 1

Solution:

P(A) = 1/2, P(B) = 2/5

Since, A and B are independent events, so

$P(A\cap B)=P(A).P(B)=\dfrac12.\dfrac25=\dfrac15$

Now, $P[(A\cap B)|A]+P[A\cap B)|B]=2P(A\cap B)$ [$\because$ Independent]

$=2.\dfrac15=\dfrac25$