problem 3

Let $S$ be event that a man in the US is a smoker and $C$ be event man has cancer. From problem conditions:

\begin{matrix} P (S) = 0.216 P (C | S) = 23 P (C | S^{c}) P (C ∣ ∣ S^{c}) = \frac{1}{23} P (C ∣ ∣ S) \end{matrix}

Lets use Bayes’ theorem

\begin{matrix} P (S | C) & = \frac{P (S) P (C | S)}{P (C)} = \frac{P (S) P (C | S)}{P (S) P (C | S) + P (S^{c}) P (C | S^{c})} = \frac{P (S) \cdot 23 P (C | S^{c})}{P (S) \cdot 23 P (C | S^{c}) + P (S^{c}) P (C | S^{c})} = \frac{P (S)}{23 P (S) + P (S^{c})} = \frac{23 \cdot 0.216}{23 \cdot 0.216 + 0.784} \approx 0.864 \end{matrix}