10 hunters shoot at 10 ducks each choosing his target unifor

10 hunters shoot at 10 ducks each choosing his target uniformly at random. Let X be the

number of ducks that escape. Find EX. (Hint: Let Xi = 1 if i-th duck escapes. What is the probability that hunter j hits duck i?)

Solution

If all hunters choose the same duck, 9 ducks can escape

At the other extreme, if each hunter select each duck separately then no duck escapes

Hence no of ducks X can take values as 0 , 1,....9

Pdf of X will be as follows:

Favourable ways for 0 = 10P10 = 10! ways

Favourable ways for x =1 = 10P9 = 10!

Favourable ways for x=2   is 10P8 = 10!/2! and so on

E(X) = 0.999999