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Statistics and Probability
Upper level probability

bingageofrey

Answered question

2022-05-30

In a survey, 70% of men and 55% of women said that they smoked. If the proportion of men to women is 60:40 and a person from the survey was chosen at random and found to be a smoker, what is the probability that this person is a woman?

Answer & Explanation

Don Sumner

Skilled2023-05-18Added 184 answers

To solve this problem, we can use Bayes' theorem to calculate the probability that the chosen person is a woman given that they are a smoker.
Let's denote the events as follows:
M = event that a person is a man
W = event that a person is a woman
S = event that a person is a smoker
We are given:

P (S | M) = 0.70

(probability that a man is a smoker)

P (S | W) = 0.55

(probability that a woman is a smoker)

P (M) = 0.60

(proportion of men in the survey)

P (W) = 0.40

(proportion of women in the survey)
We need to find

P (W | S)

, which represents the probability that a person is a woman given that they are a smoker.
According to Bayes' theorem:

P (W | S) = \frac{P (S | W) \cdot P (W)}{P (S)}

To calculate

P (S)

, we can use the law of total probability:

P (S) = P (S | M) \cdot P (M) + P (S | W) \cdot P (W)

Substituting the given values:

P (S) = 0.70 \cdot 0.60 + 0.55 \cdot 0.40

Simplifying:

P (S) = 0.42 + 0.22 = 0.64

Now, we can calculate

P (W | S)

P (W | S) = \frac{P (S | W) \cdot P (W)}{P (S)} = \frac{0.55 \cdot 0.40}{0.64}

Simplifying the expression:

P (W | S) = \frac{0.22}{0.64} = \frac{11}{32}

Therefore, the probability that a person chosen at random from the survey, who is a smoker, is a woman is

\frac{11}{32}

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