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]

To tackle this probability problem, let’s dive in and break down the different parts. (i) To find the probability that both chosen individuals have the name Lee: The total number of ways to choose 2 people from 36 is \( \binom{36}{2} = 630 \). The number of ways to choose 2 people from the 25 named Lee is \( \binom{25}{2} = 300 \). Thus, the probability is: \[ P(\text{both Lee}) = \frac{\binom{25}{2}}{\binom{36}{2}} = \frac{300}{630} = \frac{10}{21}. \] (ii) To find the probability that they are married to each other: There are 8 married couples named Lee and 3 married couples named Chan, totaling 11 married couples. The total number of ways to choose 2 people from the 36 is still 630. The number of ways to choose married couples is \( 11 \). Therefore, the probability is: \[ P(\text{married}) = \frac{11}{630}. \] (iii) To find the probability that they both have the name Lee, given that they are married to each other: Out of the 11 couples, only 8 of them have the name Lee. Hence, the conditional probability is: \[ P(\text{both Lee} | \text{married}) = \frac{8}{11}. \] (iv) To find the probability that they are a man and woman with the same name: There are 4 single men and 5 single women named Lee, and 2 single men and 3 single women named Chan. This gives a total of \( 4 \times 5 + 2 \times 3 = 26 \) combinations. Hence the probability is: \[ P(\text{man and woman same name}) = \frac{26}{630} = \frac{13}{315}. \] (v) To find the probability that they are married to each other, given that they are a man and a woman with the same name: There are 8 married couples named Lee and 3 couples named Chan, totaling 11 couples. There are 26 man and woman combinations with the same name. Thus: \[ P(\text{married} | \text{man and woman same name}) = \frac{11}{26}. \] Now you've got all the probabilities in a neat little bundle! Good luck with your gathering analysis!