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
\(\frac{13}{15}\)
Verified Correct Answer: D.
Step-by-Step Mathematical Solution
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)
\(P = (\frac{2}{3} \times \frac{2}{5}) + (\frac{2}{5} \times \frac{1}{3}) + (\frac{2}{3} \times \frac{3}{5})\)
= \(\frac{4}{15} + \frac{3}{15} + \frac{6}{15}\)
= \(\frac{13}{15}\)
Source: Myschool Past Questions Archive (Record #27399)