Question

The probabilities of a husband and his wife surviving for 20 more years are 0.8 and 0.9 respectively. Find the probability that after 20 years: a. both of them are alive b. at least one of them is alive.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{A}\mathrm{be}\mathrm{the}\mathrm{event}\mathrm{that}\mathrm{a}\mathrm{husband}\mathrm{will}\mathrm{be}\mathrm{alive}\mathrm{after}20\mathrm{yrs}. \\ \mathrm{Let}\mathrm{B}\mathrm{be}\mathrm{the}\mathrm{event}\mathrm{that}\mathrm{a}\mathrm{wife}\mathrm{will}\mathrm{be}\mathrm{alive}\mathrm{after}20\mathrm{yrs}. \\ \mathrm{Now},\mathrm{P}\left(\mathrm{A}\right)=0.8;\mathrm{P}\left(\mathrm{B}\right)=0.9 \\ \mathrm{Now},\mathrm{P}\left(\overline{\mathrm{A}}\right)=1-\mathrm{P}\left(\mathrm{A}\right)=1-0.8=0.2 \\ \mathrm{Now},\mathrm{P}\left(\overline{\mathrm{B}}\right)1-\mathrm{P}\left(\mathrm{B}\right)=1-0.9=0.1 \\ \left(\mathrm{a}\right). \\ \mathrm{Probability}\mathrm{that}\mathrm{both}\mathrm{of}\mathrm{them}\mathrm{are}\mathrm{alive}=\mathrm{P}\left(\mathrm{A}\right).\mathrm{P}\left(\mathrm{B}\right)=0.8\times 0.9=0.72 \\ \left(\mathrm{b}\right). \\ \mathrm{Probability}\mathrm{that}\mathrm{none}\mathrm{of}\mathrm{them}\mathrm{is}\mathrm{alive}=\mathrm{P}\left(\overline{\mathrm{A}}\right).\mathrm{P}\left(\overline{\mathrm{B}}\right)=0.2\times 0.1=0.02 \\ \mathrm{Probability}\mathrm{that}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{of}\mathrm{them}\mathrm{is}\mathrm{alive}=1-0.02=0.08 \end{gathered}$$