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

$\frac{2}{15}$

$\frac{3}{15}$

$\frac{7}{15}$

$\frac{13}{15}$

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)

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

= $\frac{4}{15} + \frac{3}{15} + \frac{6}{15}$

= $\frac{13}{15}$

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