A die is rolled 4 times Let X and Y be the number of appeara

A die is rolled 4 times. Let X and Y be the number of appearances of 1 and 6 respectively. Find the joint probability mass function of (X, Y).

Solution

x - 1 appears

Y - 6 appears

P(X,y) = Both 1 and 6 appear

x = 0 1 2 3 4

Y = 0 1 2 3 4

P(X=r, y=s) = 6Cr(0.1667)r(0.8333)6-r 6Cs(0.1667)s(0.8333)6-s

= 6Cr6Cs(0.1667)r+s(0.8333)12-r-s, x=0,1,2,3,4 and y =0,1,2,3,4