Sleeping arrangements.

• Your family is trying to decide sleeping arrangements in the house. There are 10 young people staying in the house who will be divided into two rooms. You will choose a set of 5 people to sleep in the first room and the rest will sleep in the second room. A sleeping arrangement is acceptable if it meets the following two criteria:1) Your two youngest cousins (say, 7 and 8 years old) always fight, so exactly one of them must be in the first room. 2) There are 4 people over 18 years old, and exactly 2 of them must be in the first room to supervise the others.

You are too lazy to work out the arrangement by hand, so you decide to do randomly by choosing a set of 5 people at random to sleep in the first room, and repeat this until you get an acceptable arrangement.

a. For a single draw of 5 people, what is the probability that you will draw an acceptable sleeping arrangement?

b. What is the expected number of draws of 5 people until you get an acceptable arrangement? Note: Each drawing of 5 people is independent of every other drawing of 5 people.