Question

Two balls are chosen randomly from an urn containing 9 yellow, 5 blue, and 3 magenta balls. Suppose that we win $3 for each blue ball selected, we lose $2 for each yellow ball selected and we win $0 for each magenta ball selected. Let X denote our winnings. What are the possible values of X, what are the probabilities associated with each value (i.e., find the probability mass function of X), and what is the expectation value of X,E[X]?

Answer 1

here possible values of X are -4 ,-2,0,1,3,6

probabilities associated with each value are as follows:

P(X=-4)=P(both are yellow) =(9/17)*(8/16)=9/34

P(X=-2)=P(1 yellow and other magneta)=2*(9/17)*(3/16)=27/136

P(X=0)=P(both magneta)=(3/17)*(2/16)=3/136

P(X=1)=P(1 yellow and other blue) =2*(9/17)*(5/16)=45/136

P(X=3)=P(1 magneta and other blue) =2*(5/17)*(3/16)=15/136

P(X=6) =P(both blue) =(5/17)*(4/16)=5/68

expected value

E(X)=xP(x) =-4*(9/34)-2*(27/136)+0*(3/136)+1*(45/136)+3*(15/136)+6*(5/68)= -0.35