Question

In a cosmopolitan city, the population comprises of 30% female and 70% male. Suppose that 5% of females and 30% of males in the population are foreigners. A person is selected at random from this population. Given that the selected person is a foreigner, the probability that the person is a female is ________ (round off to three decimal places).

Hint:

Bayes' Theorem is useful for finding conditional probabilities, especially when given partial information about different subsets of a population.

Answer 1

Correct Answer: 0.066

This is a problem of conditional probability. We can use Bayes' Theorem to calculate the probability that the selected person is female, given that they are a foreigner. Let:
- \( P(F) \) be the probability of selecting a female = 0.30,
- \( P(M) \) be the probability of selecting a male = 0.70,
- \( P(F_{\text{for}}) \) be the probability of selecting a female foreigner = 0.05 \times 0.30 = 0.015,
- \( P(M_{\text{for}}) \) be the probability of selecting a male foreigner = 0.30 \times 0.70 = 0.21.
The total probability of selecting a foreigner, \( P(\text{for}) \), is the sum of the probabilities of selecting a female or male foreigner: \[ P(\text{for}) = P(F_{\text{for}}) + P(M_{\text{for}}) = 0.015 + 0.21 = 0.225 \] Now, using Bayes' Theorem, the probability that the selected person is female, given that they are a foreigner, is: \[ P(F \mid \text{for}) = \frac{P(F_{\text{for}})}{P(\text{for})} = \frac{0.015}{0.225} = 0.0667 \] Thus, the probability that the selected person is a female is: \[ \boxed{0.067} \]