9. [In this question, give each of your answers cither as an exact fraction in its lowest terms or as a decimal correct to 3 places.] There are 36 people at a gathering of two families. There are 25 pcople with the name Lee and 11 people with the name Chan. Of the 25 people named lee, 4 are single men, 5 are single women and there are 8 married couples. Of the 11 people named Chan, 2 are single men, 3 are single women and there are 3 married couples. Two people are chosen at random from the gathering. (i) Show that the probability that they both have the name Lee is \( \frac{10}{21} \). (ii) Find the probability that they are married to each other. (iii) Find the probability that they both have the name Lee, given that they are married to each other. (iv) Find the probability that they are a man and woman with the same name. (v) Find the probability they are married to each other, given that they are a man and a woman with the same name. [N1997/IV6]

Let's break down the problem step by step, extracting the known conditions and calculating the required probabilities. ### Known Conditions: - Total people = 36 - People named Lee = 25 - Single men = 4 - Single women = 5 - Married couples = 8 (which means 16 people) - People named Chan = 11 - Single men = 2 - Single women = 3 - Married couples = 3 (which means 6 people) ### (i) Probability that both have the name Lee To find the probability that both chosen people have the name Lee, we can use the formula for probability: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \] **Total outcomes** when choosing 2 people from 36: \[ \text{Total outcomes} = \binom{36}{2} = \frac{36 \times 35}{2} = 630 \] **Favorable outcomes** when both have the name Lee: \[ \text{Favorable outcomes} = \binom{25}{2} = \frac{25 \times 24}{2} = 300 \] Now, we can calculate the probability: \[ P(\text{both Lee}) = \frac{300}{630} = \frac{10}{21} \] ### (ii) Probability that they are married to each other **Total married couples**: - Lee: 8 couples - Chan: 3 couples - Total married couples = 8 + 3 = 11 **Favorable outcomes** when choosing 2 people who are married to each other: \[ \text{Favorable outcomes} = 11 \] Now, we can calculate the probability: \[ P(\text{married}) = \frac{11}{630} \] ### (iii) Probability that they both have the name Lee, given that they are married to each other To find this conditional probability, we use: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Where: - \(A\) = both have the name Lee - \(B\) = they are married to each other **Favorable outcomes for \(A \cap B\)** (both are married Lee couples): \[ \text{Favorable outcomes} = 8 \] Now, we can calculate: \[ P(A|B) = \frac{8/630}{11/630} = \frac{8}{11} \] ### (iv) Probability that they are a man and woman with the same name **Favorable outcomes**: - Lee: 4 single men and 5 single women = 4 * 5 = 20 - Chan: 2 single men and 3 single women = 2 * 3 = 6 - Total favorable outcomes = 20 + 6 = 26 Now, we can calculate the probability: \[ P(\text{man and woman same name}) = \frac{26}{630} = \frac{13}{315} \] ### (v) Probability they are married to each other, given that they are a man and a woman with the same name Using the conditional probability formula again: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Where: - \(A\) = they are married to each other - \(B\) = they are a man and woman with the same name **Favorable outcomes for \(A \cap B\)** (married couples who are a man and woman with the same name): - Lee: 8 couples - Chan: 3 couples - Total = 11 **Total outcomes for \(B\)** (man and woman with the same name) = 26 (as calculated previously). Now, we can calculate: \[ P(A|B) = \frac{11/630}{26/630} = \frac{11}{26} \] ### Summary of Answers: (i) \( \frac{10}{21} \) (ii) \( \frac{11}{630} \) (iii) \( \frac{8}{11} \) (iv) \( \frac{13}{315} \) (v) \( \frac{11}{26} \)