Questions: If one of the 1156 people is randomly selected, find the probability that the person is a man or a heavy smoker.

Solution Steps

To find the probability that a randomly selected person is a man or a heavy smoker, we need to use the principle of inclusion-exclusion. This principle states that the probability of either event A or event B occurring is the sum of the probabilities of each event minus the probability of both events occurring.

1. Calculate the probability of selecting a man.
2. Calculate the probability of selecting a heavy smoker.
3. Calculate the probability of selecting a person who is both a man and a heavy smoker.
4. Use the principle of inclusion-exclusion to find the probability of selecting a person who is either a man or a heavy smoker.

The probability of selecting a man from the group is given by:

\[ P(\text{Man}) = \frac{\text{Number of Men}}{\text{Total People}} = \frac{586}{1156} \approx 0.5069 \]

The probability of selecting a heavy smoker is calculated as follows:

\[ P(\text{Heavy Smoker}) = \frac{\text{Number of Heavy Smokers}}{\text{Total People}} = \frac{82}{1156} \approx 0.07093 \]

The probability of selecting a person who is both a man and a heavy smoker is:

\[ P(\text{Man and Heavy Smoker}) = \frac{\text{Number of Men and Heavy Smokers}}{\text{Total People}} = \frac{42}{1156} \approx 0.03633 \]

Using the principle of inclusion-exclusion, we find the probability of selecting a person who is either a man or a heavy smoker:

\[ P(\text{Man or Heavy Smoker}) = P(\text{Man}) + P(\text{Heavy Smoker}) - P(\text{Man and Heavy Smoker}) \]

Substituting the values we calculated:

\[ P(\text{Man or Heavy Smoker}) \approx 0.5069 + 0.07093 - 0.03633 \approx 0.5415 \]

Final Answer