The probabilities that a man and his wife live for 80 years are \(\frac{2}{3}\) and \(\frac{3}{5}\) respectively. Find the probability that at least one of them will live up to 80 years

A.

\(\frac{2}{15}\)

B.

\(\frac{3}{15}\)

C.

\(\frac{7}{15}\)

D.

\(\frac{13}{15}\)

Correct answer is D

Man lives = \(\frac{2}{3}\) not live = \(\frac{1}{3}\)

Wife lives = \(\frac{3}{5}\) not live = \(\frac{2}{5}\)

P(at least one lives to 80 years) = P(man lives to 80 not woman) + P(woman lives to 80 and not man) + P(both live to 80)

\(P = (\frac{2}{3} \times \frac{2}{5}) + (\frac{2}{5} \times \frac{1}{3}) + (\frac{2}{3} \times \frac{3}{5})\)

= \(\frac{4}{15} + \frac{3}{15} + \frac{6}{15}\)

= \(\frac{13}{15}\)