Suppose the integers 15 are randomly assigned to five people

Suppose the integers 1-5 are randomly assigned to five people(call them A,B,C,D,and E). Them they play a game in which A and B first compare their numbers and the person with the higher number wins the round, then that person proceeds to compare his or her number with C\'s number. The game proceeds in this manner until the fourth comparison (involving E and the winner of the previous comparison) takes place. Let the random variable W represent the number of comparisons for which A is the Winner. Determine the probability distribution of W.

Solution

If A has number 1 there are no ways in which he can win

If A has number 2 then there is only one way in which he can win

If A has number 3 then he can win if B and C have numbers 1 and 2. This can be done in 2! = 2 ways

If A has number 4 then he can win if B, C and D have numbers 1,2 or 3. This can be done in 3! ways = 6 ways

If A has number 5 then A will always be greater than B,C,D or E. Number of ways in which the other numbers can be arranged is 4! = 24

Total number of ways = 1 + 2 + 6 + 24 = 33

Total number of ways in which the 5 numbers can be arranged is 5! = 120

Probability = 33/120 = 0.275