Question

1. Suppose there are ? men including Bob and John. If we randomly arrange them in a line, what is the probability that exactly ? men stand between Bob and John?

2. If ? and ? are independent, please show that: (30%)

(a) ? and ?c are independent

(b) ?c and ?c are independent

(c)Foragiven ?, if ?, ? and ? aremutuallyindependent, ?(?∩?)≠0 aswell

as ?(?) ≠ 0, please show that ?(?|? ∩ ?) = ?(?)

Answer 1

1;

Total number of ways n men can arrange themselves in a line is n!.

Now suppose there are exactly k men between Bob and Jhon. Let us assume Bob, k men, and then Jhon as one man. So number of men remaining is n - k -2. These n-k-2 men can arrange themselves in (n-k-2)! ways, Rest k men can arrange themselves in k! ways so possible number of ways exactly ? men stand between Bob and John is

(n-k-2)! * k!

The probability that exactly ? men stand between Bob and John is

P = [ (n-k-2)! * k! ] / n!