Question

In orange country,51%of the adults are males.one adult is rendomly
Selected for a survey involvingcredit card usage.

Answer 1

Answer on Question #71803 – Math – Statistics and Probability

Question

In Orange County, 51% of the adults are males. One adult is randomly selected for a survey involving credit card usage.

a) Find the prior probability that the selected person is a female.

b) It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of females smoke cigars. Use this additional information to find the probability that the selected subject is a female.

Solution

Let's use the following notation:

$$\begin{array}{l} M = \text{male} \\ \overline{M} = \text{female (or not male)} \\ C = \text{cigar smoker} \\ \overline{C} = \text{not cigar smoker} \\ \end{array}$$

a) A prior probability is an initial probability value originally obtained before any additional information is obtained.

The prior probability that the selected person is a female

$$P(\overline{M}) = 1 - P(M) = 1 - 0.51 = 0.49$$

b) Bayes' Theorem

The probability of event $A$, given that event $B$ has subsequently occurred, is

$$P(A|B) = \frac{P(A)P(B|A)}{[P(A)P(B|A)] + [P(\overline{A})P(B|\overline{A})]}$$

Based on the additional information:

$$\begin{array}{l} P(M) = 0.51 \\ P(\overline{M}) = 0.49 \\ P(C|M) = 0.095 \\ P(C|\overline{M}) = 0.017 \\ \end{array}$$

We can now apply Bayes' Theorem.

Then the probability that the selected subject is female using additional information that is later obtained

$$\begin{array}{l} P(\overline{M}|C) = \frac{P(\overline{M})P(C|\overline{M})}{\left[P(\overline{M})P(C|\overline{M})\right] + \left[P(M)P(C|M)\right]} \\ P(\overline{M}|C) = \frac{0.49(0.017)}{[0.49(0.017)] + [0.51(0.095)]} \approx 0.1467 \\ \end{array}$$

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