Let A and B be two tests performed by a randomized algorithm

Let A and   B be two tests performed by a randomized algorithm, each of which either succeeds or fails. The two tests need not be independent. Let the probability of failure for A be 0.5, and for B be0.2. Which of the following values could possibly be the probability that both tests fail?

Solution

Assume A failure = P(A)=0.5

Assume B failure = P(B)=0.2

The Probability of A & B fails =P(A and B) =P(A).P(B/A) (more data needed ,if both events are dependent to calculate )

The Probability of A & B fails =P(A and B) =P(A).P(B)=0.5 * 0.2 = 0.1 (if both events are independent)