1. **Problem Statement:** We have two problems involving probability based on frequency tables. **Problem 3:** Estimate the likelihood that: a) there is a four day gap between refills b) there is at least a four day gap between refills. **Problem 4:** Given the age distribution of prison inmates, find probabilities: a) the prisoner was male b) the prisoner was aged between 17 and 19 c) the prisoner was 19 or under given that the prisoner was female d) the prisoner was 19 or under given that the prisoner was male e) the prisoner was female given that the prisoner was aged 60+. --- 2. **Formulas and Rules:** - Probability of an event $E$ is $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. - For conditional probability $P(A|B) = \frac{P(A \cap B)}{P(B)}$. - "At least" means greater than or equal to. --- 3. **Problem 3 Calculations:** - Total frequency = $37 + 81 + 48 + 17 + 6 + 1 = 190$. (a) Probability of exactly 4 day gap: $$P(4) = \frac{17}{190}$$ (b) Probability of at least 4 day gap means days 4, 5, or 6: $$P(\geq 4) = \frac{17 + 6 + 1}{190} = \frac{24}{190}$$ --- 4. **Problem 4 Calculations:** - Total inmates = 5038 - Total females = 216 - Total males = 4822 (a) Probability prisoner was male: $$P(\text{male}) = \frac{4822}{5038}$$ (b) Probability prisoner aged 17-19: $$P(17\text{-}19) = \frac{448}{5038}$$ (c) Probability prisoner was 19 or under given female: - Females aged 15,16,17-19 = $0 + 5 + 26 = 31$ - Total females = 216 $$P(\leq 19 | \text{female}) = \frac{31}{216}$$ (d) Probability prisoner was 19 or under given male: - Males aged 15,16,17-19 = $6 + 23 + 422 = 451$ - Total males = 4822 $$P(\leq 19 | \text{male}) = \frac{451}{4822}$$ (e) Probability prisoner was female given aged 60+: - Females aged 60+ = 5 - Total aged 60+ = 153 $$P(\text{female} | 60+) = \frac{5}{153}$$ --- 5. **Final answers:** - 3a) $\frac{17}{190} \approx 0.0895$ - 3b) $\frac{24}{190} \approx 0.1263$ - 4a) $\frac{4822}{5038} \approx 0.9573$ - 4b) $\frac{448}{5038} \approx 0.0889$ - 4c) $\frac{31}{216} \approx 0.1435$ - 4d) $\frac{451}{4822} \approx 0.0935$ - 4e) $\frac{5}{153} \approx 0.0327$