N1997 P2 Q6

There are \(36\) people at a gathering of two families. There are \(25\) people 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.

1. Show that the probability that they both have the name Lee is \(\frac{10}{21}\).
2. Find the probability that they are married to each other.
3. Find the probability that they both have the name Lee, given that they are married to each other.
4. Find the probability that they are a man and a woman with the same name.
5. Find the probability that they are married to each other, given that they are a man and a woman with the same name.