Suppose it costs 3 to play a game where you select 3 cards w

Suppose it costs $3 to play a game where you select 3 cards (without replacement) from a standard deck of cards (52 cards), and for every club (13 in deck) you select, you will be paid $2. Let X represent the number of clubs selected while playing this game. Let Y be the amount of money you win/lose. Find the PMF for Y (in terms of y, not specified as probabilities). What is the expected value of Y? Interpret this value.

Solution

X can take values 0,1,2,3.

P( X = 0) = 39C3 * 13C0/52C3 = 0.4135

P( X = 1) = 39C2 * 13C1/52C3 = 0.4359

P( X = 2) = 39C1 * 13C2/52C3 = 0.1377

P( X = 3) = 39C0 * 13C3/52C3 = 0.0129

Now, when X = 0, Y = 0-3 =-3

when X = 1, Y = 2-3 =-1

when X = 2, Y = 4-3 = 1

when X = 3, Y = 6-3 = 3

So, pmf of Y is

P(Y = -3) = 0.4135

P(Y = -1) = 0.4359

P(Y = 1) = .0.1377

P(Y = 3) = 0.0129

b)

Expected value of Y = -3*0.4135-1*0.4359+1*0.1377+3*0.0129 = -1.5

So, if the game goes on long term, I will expect to lose $1.5