I need help with these counting/probability questions. 3. The following question
ID: 3363812 • Letter: I
Question
I need help with these counting/probability questions.
3. The following questions are not related. (a) (15 pts) The 6 h at experiment is performed with three married couples. . Find the probability of A, derangement (nobody gets their own hat), of B, the event that own her own husband's hat, and of C, the event that each man got his wife's hat. Are 4 and B independent? Are B and C independent? Are they pairwise independent? Explain (o) (i0 pts) The US senate has 100 senators two from each state - and 21 of them are women. In Fact in three states (Cal, N. Hamp, Wash,) both of the senators are women and in fteen other states, ONE of the two senators is a woman. i. At lunch time all 100 senators were lined up "at random" to form a waiting line to get a seat for lunch. Carefully describe the sample space for this experiment and give its size. i. What is the probabilty that the first and last senators in the line are women? Explain. iüi. What is the probability that the two senators from each state are adjacent in the line? Explain your answer.Explanation / Answer
b)
i) Here the sample space is the US senate of size 100.
ii) Two women for the first and last position in the queue can be selected in 21C2 ways. And these two women can be arranged among themseleves in 2! ways.
Remaining 98 senators can be arranged in 98! ways.
Hence total possible ways arrangement such that first and last senators in the line are women = 21C2 * 2! * 98!
Total possible arrangements are = 100!
Required probability = (21C2 * 2! * 98!)/100! = 21C2*2/(99*100) = 0.0424
iii)
If 2 senators from each state has adjacent to be other, there are 50!*2! possible ways of doing this.
Required probability = 50!*2!/100!