Question

Three balls are randomly chosen from an urn containing 3 white, 3 red, and 5 blackballs. Suppose that we win $1 for each white ball selected, and lose $1 for each red ball selected. If X denotes our total winnings from the experiment, then:

a) What values can X take?

b) What is the PMF of X?

c) Show that this is a valid PMF.

Answer 1

a)

since player can pick 3 white or 3 black balls , therefore the values

X can take are -3,-2,-1,0,1,2,3

b)

below is pmf of X:

P(X=-3) =P(all three are red) =(3C3)/(11C3) =1/165

P(X=-2) =P(2 red and 1 black)=(3C2)*(5C1)/(11C3)=3*5/165 =15/165

P(X=-1)=P(2 red and 1 white)+P(1 red and 2 black)=(3C2)*(3C1)/(11C3)+(3C1)*(5C2)/(11C3)

=39/165

P(X=0)=P(3 black)+P(1red,1 white, 1 black)=(5C3)/(11C3)+(3C1)*(3C1)*(5C1)/(11C3)=55/165

P(X=1)=P(1 red and 2 white)+P(1 white and 2 black)=39/165

P(X=2)=P(2 white and 1 black)=15/165

P(X=3)=P(all three are white) =(3C3)/(11C3) =1/165

c)

since each of individual probability is between 0 and 1

and sum of all probability points is 1 , therefore this is a valid pmf.