Questions: Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted. The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker Light smoker Heavy smoker Total ------------------------------------------------------ Men 315 54 72 441 Women 308 86 93 487 Total 623 140 165 928 If one of the 928 subjects is randomly selected, find the probability that the person chosen is a woman given that the person is a light smoker. Round to nearest thousandth.

Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted. The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker Light smoker Heavy smoker Total ------------------------------------------------------ Men 315 54 72 441 Women 308 86 93 487 Total 623 140 165 928 If one of the 928 subjects is randomly selected, find the probability that the person chosen is a woman given that the person is a light smoker. Round to nearest thousandth.
Transcript text: Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted. The table below describes the smoking habits of a group of asthma sufferers. | | Nonsmoker | Light smoker | Heavy smoker | Total | |--------|-----------|--------------|--------------|-------| | Men | 315 | 54 | 72 | 441 | | Women | 308 | 86 | 93 | 487 | | Total | 623 | 140 | 165 | 928 | If one of the 928 subjects is randomly selected, find the probability that the person chosen is a woman given that the person is a light smoker. Round to nearest thousandth.
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Solution

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Solution Steps

To find the probability that a randomly selected person is a woman given that the person is a light smoker, we use conditional probability. The formula for conditional probability is P(A|B) = P(A and B) / P(B). Here, A is the event that the person is a woman, and B is the event that the person is a light smoker. We need to find the number of women who are light smokers and divide it by the total number of light smokers.

Step 1: Identify Events

Let \( A \) be the event that the person chosen is a woman, and let \( B \) be the event that the person is a light smoker. We need to find \( P(A|B) \).

Step 2: Use Conditional Probability Formula

The formula for conditional probability is given by:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Step 3: Calculate \( P(A \cap B) \) and \( P(B) \)

From the table, we have:

  • \( P(A \cap B) \) (the probability of being a woman and a light smoker) is represented by the number of women who are light smokers, which is \( 86 \).
  • \( P(B) \) (the total number of light smokers) is \( 140 \).
Step 4: Calculate the Probability

Now we can substitute the values into the formula:

\[ P(A|B) = \frac{86}{140} \approx 0.6142857142857143 \]

Step 5: Round the Result

Rounding \( 0.6142857142857143 \) to four significant digits gives us \( 0.614 \).

Final Answer

The probability that the person chosen is a woman given that the person is a light smoker is \\(\boxed{0.614}\\).

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