Probability Calculation for Men's Health Data

This problem involves calculating the joint probability of two events (being a smoker and having cancer) using the information provided about a population of men.

Given Probabilities

• Let C be the event that a man suffers from cancer. Given: $P(C) = 3\% = 0.03$
• Let S be the event that a man is a smoker. Given: $P(S) = 15\% = 0.15$
• Let S U C be the event that a man is either a smoker or has cancer (or both). Given: $P(S \cup C) = 10\% = 0.10$

Finding Joint Probability (Smoker AND Cancer)

We need to find the probability that a randomly selected man is both a smoker and suffers from cancer, denoted as $P(S \cap C)$.

We use the addition rule for probabilities:

$P(S \cup C) = P(S) + P(C) - P(S \cap C)$

To find $P(S \cap C)$, we rearrange the formula:

$P(S \cap C) = P(S) + P(C) - P(S \cup C)$

Calculation Steps

1. Substitute the given values into the rearranged formula:
   $P(S \cap C) = 0.15 + 0.03 - 0.10$
2. Perform the addition:
   $P(S \cap C) = 0.18 - 0.10$
3. Perform the subtraction to find the final probability:
   $P(S \cap C) = 0.08$

Therefore, the probability that a randomly selected man is both a smoker and suffering from cancer is 0.08.