In the game of redball two drawings are made without replacements from a bowl that has four white ping pong balls and two red ping pong balls. The amount won is determined by how many ping-pong balls. The amount won is determined by how many of the red balls are selected. For a $5 bet, a player can opt to be paid under either Rule A or Rule B, as shown. If you were playing the game, which would you choose? Why?
For A: # of Red ball drawns 0,1,2 and their payoff in dollars 0 , $2, $10 respectively.
Similarly for B: # of Red ball drawns 0,1,2, and their payoffs in dollars 0, $1, $20.
Solution: Let X be a discrete variable with probability p_x(k). The expected value of X, denoted by E(x) is given by E(x) = (Sum of for all k) k*p_x(k).
For Rule A: we have from the hypergeometric random variable P(X=k) = p_x(k) = [2_C_k * 4_C_(2-k)]/6_C_2, for k = 0,1,2.
THus, E(x) = (Sum from 0 to 2) K*[2_C_k * 4_C_(2-k)]/6_C_2 = 0* [2_C_0 * 4_C_(2-0)]/6_C_2 + 1* [2_C_1 * 4_C_(2-1)]/6_C_2 + 2* [2_C_2 * 4_C_(2-2)]/6_C_2 = 0 + 2*4/15 + 2*6/15 = 20/15.
Can someone please help me? I don't know how to us the payoffs. I don't know where would I plug the payoffs in the equations. Anything would be appreciate it. Thank you.