Answer :
To determine the probability that a randomly selected person is either a man or a heavy smoker, we can break down the problem step by step.
### Step 1: Calculate the probability of being a man
First, note the total number of people is 1064, and the number of men is 533. The probability that a randomly selected person is a man ([tex]\( P(\text{Man}) \)[/tex]) is given by:
[tex]\[ P(\text{Man}) = \frac{\text{Number of men}}{\text{Total number of people}} = \frac{533}{1064} \approx 0.5009 \][/tex]
### Step 2: Calculate the probability of being a heavy smoker
Similarly, we know there are 89 heavy smokers out of a total of 1064 people. The probability that a randomly selected person is a heavy smoker ([tex]\( P(\text{Heavy Smoker}) \)[/tex]) is given by:
[tex]\[ P(\text{Heavy Smoker}) = \frac{\text{Number of heavy smokers}}{\text{Total number of people}} = \frac{89}{1064} \approx 0.0836 \][/tex]
### Step 3: Calculate the probability of being both a man and a heavy smoker
From the table, 42 people are both men and heavy smokers. The probability that a randomly selected person is both a man and a heavy smoker ([tex]\( P(\text{Man and Heavy Smoker}) \)[/tex]) is given by:
[tex]\[ P(\text{Man and Heavy Smoker}) = \frac{\text{Number of men who are heavy smokers}}{\text{Total number of people}} = \frac{42}{1064} \approx 0.0395 \][/tex]
### Step 4: Apply the principle of inclusion and exclusion
The probability of being either a man or a heavy smoker ([tex]\( P(\text{Man or Heavy Smoker}) \)[/tex]) can be found using the principle of inclusion and exclusion:
[tex]\[ P(\text{Man or Heavy Smoker}) = P(\text{Man}) + P(\text{Heavy Smoker}) - P(\text{Man and Heavy Smoker}) \][/tex]
Substituting the values obtained:
[tex]\[ P(\text{Man or Heavy Smoker}) = 0.5009 + 0.0836 - 0.0395 \approx 0.5451 \][/tex]
Thus, the probability that a randomly selected person is either a man or a heavy smoker is approximately 0.545. Therefore, the correct answer is:
[tex]\[ \boxed{0.545} \][/tex]
### Step 1: Calculate the probability of being a man
First, note the total number of people is 1064, and the number of men is 533. The probability that a randomly selected person is a man ([tex]\( P(\text{Man}) \)[/tex]) is given by:
[tex]\[ P(\text{Man}) = \frac{\text{Number of men}}{\text{Total number of people}} = \frac{533}{1064} \approx 0.5009 \][/tex]
### Step 2: Calculate the probability of being a heavy smoker
Similarly, we know there are 89 heavy smokers out of a total of 1064 people. The probability that a randomly selected person is a heavy smoker ([tex]\( P(\text{Heavy Smoker}) \)[/tex]) is given by:
[tex]\[ P(\text{Heavy Smoker}) = \frac{\text{Number of heavy smokers}}{\text{Total number of people}} = \frac{89}{1064} \approx 0.0836 \][/tex]
### Step 3: Calculate the probability of being both a man and a heavy smoker
From the table, 42 people are both men and heavy smokers. The probability that a randomly selected person is both a man and a heavy smoker ([tex]\( P(\text{Man and Heavy Smoker}) \)[/tex]) is given by:
[tex]\[ P(\text{Man and Heavy Smoker}) = \frac{\text{Number of men who are heavy smokers}}{\text{Total number of people}} = \frac{42}{1064} \approx 0.0395 \][/tex]
### Step 4: Apply the principle of inclusion and exclusion
The probability of being either a man or a heavy smoker ([tex]\( P(\text{Man or Heavy Smoker}) \)[/tex]) can be found using the principle of inclusion and exclusion:
[tex]\[ P(\text{Man or Heavy Smoker}) = P(\text{Man}) + P(\text{Heavy Smoker}) - P(\text{Man and Heavy Smoker}) \][/tex]
Substituting the values obtained:
[tex]\[ P(\text{Man or Heavy Smoker}) = 0.5009 + 0.0836 - 0.0395 \approx 0.5451 \][/tex]
Thus, the probability that a randomly selected person is either a man or a heavy smoker is approximately 0.545. Therefore, the correct answer is:
[tex]\[ \boxed{0.545} \][/tex]
