The probability of an event P happening is 1/5 and that of event Q is 1/4 . If the events are independent…

Analyze the question: We need the probability that neither P nor Q happens, given that the events are independent.

Concept: If P and Q are independent, then their complements (P not happening, Q not happening) are also independent. So:

P(neither) = P(P′) × P(Q′) = (1 − P(P)) × (1 − P(Q))

Solution:

• P(P′) = 1 − 1/5 = 4/5
• P(Q′) = 1 − 1/4 = 3/4

P(neither) = 4/5 × 3/4 = 12/20 = 3/5

Check using another method: P(P or Q) = P(P) + P(Q) − P(P)P(Q) = 1/5 + 1/4 − 1/20 = 4/20 + 5/20 − 1/20 = 8/20 = 2/5 P(neither) = 1 − 2/5 = 3/5 ✓

Answer: The probability is 3/5, which is option C.

Tip: For "neither" with independent events, find the "not happening" probability of each event (1 − p) and multiply. Note that option A (4/5) is just P(P′) and option B (3/4) is just P(Q′), so be careful not to stop halfway.