A box contains 5 red and yellow balls A ball is taken from t

A box contains 5 red and yellow balls. A ball is taken from the box and then returned to the box together with two other balls of the same color then the second ball is taken from the box. Let x be the number of the red balls from the selected during two draws.

Find the probability    mass function x

Find the cumulative distribution of x

Compute E(x) and V(x)

Solution

X -- the number of red balls selected during two draws

X can take values as 0,1 or 2.

P(x=0) = 0.5(0.5) = 0.25.

P(X=1) = P(either first ball is red and second ball non red ) or P(first ball non red and second ball red)

= 0.5(5/12)+(0.5)(7/12) = 0.1666+0.2917 =0.4583

P(X=2) = P(Both balls red) = P(First ball red and second ball red) = 0.5(7/12) = 0.2917

This is given in table as

Mean of X = 1.0417

Var (x) = 1.6251-1.0851 = 0.5400