SOLUTION: A polling organization randomly selects an adult American for a survey about credit card usage. Suppose 48% of the population is male. Suppose 9.8% of males smoke cigars and 1.52%

Question 1170691: A polling organization randomly selects an adult American for a survey about credit card usage. Suppose 48% of the population is male. Suppose 9.8% of males smoke cigars and 1.52% of females smoke cigars. If an adult is randomly selected for this survey who does not smoke cigars, what is the probability that person is a female?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's solve this problem using conditional probability and Bayes' Theorem.
**1. Define Events**
* M: The person is male.
* F: The person is female.
* C: The person smokes cigars.
* ¬C: The person does not smoke cigars.
**2. Given Probabilities**
* P(M) = 0.48 (Probability of being male)
* P(F) = 1 - P(M) = 1 - 0.48 = 0.52 (Probability of being female)
* P(C | M) = 0.098 (Probability of smoking cigars given the person is male)
* P(C | F) = 0.0152 (Probability of smoking cigars given the person is female)
**3. Calculate Probabilities of Not Smoking Cigars**
* P(¬C | M) = 1 - P(C | M) = 1 - 0.098 = 0.902 (Probability of not smoking cigars given the person is male)
* P(¬C | F) = 1 - P(C | F) = 1 - 0.0152 = 0.9848 (Probability of not smoking cigars given the person is female)
**4. Find the Probability of Not Smoking Cigars (P(¬C))**
We can use the law of total probability:
P(¬C) = P(¬C | M) * P(M) + P(¬C | F) * P(F)
P(¬C) = (0.902 * 0.48) + (0.9848 * 0.52)
P(¬C) = 0.43296 + 0.512096
P(¬C) = 0.945056
**5. Find the Probability of Being Female Given Not Smoking Cigars (P(F | ¬C))**
We use Bayes' Theorem:
P(F | ¬C) = [P(¬C | F) * P(F)] / P(¬C)
P(F | ¬C) = (0.9848 * 0.52) / 0.945056
P(F | ¬C) = 0.512096 / 0.945056
P(F | ¬C) ≈ 0.54185
**6. Round to a Reasonable Number of Decimal Places**
P(F | ¬C) ≈ 0.5419
**Answer:**
The probability that the person is female given they do not smoke cigars is approximately 0.5419.

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