1. **State Bayes’ theorem:** Bayes’ theorem relates conditional probabilities and is stated as: $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ where $P(A|B)$ is the probability of event A given B has occurred. 2. **Problem:** Given the survey data: | Gender | Yes | No | Total | |--------|-----|----|-------| | Male | 32 | 18 | 50 | | Female | 8 | 42 | 50 | | Total | 40 | 60 | 100 | Find: (1) Probability respondent answered yes given female. (2) Probability respondent was male given answered no. 3. **Formulas:** - Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ 4. **Calculations:** (1) $P(\text{Yes} | \text{Female}) = \frac{\text{Number of females who said yes}}{\text{Total females}} = \frac{8}{50} = 0.16$ (2) $P(\text{Male} | \text{No}) = \frac{\text{Number of males who said no}}{\text{Total no}} = \frac{18}{60} = 0.3$ **Final answers:** (1) $0.16$ (2) $0.3$