Milazzo 14. You are charged with the task of designing a fair, even-money card g

Question

Milazzo 14. You are charged with the task of designing a fair, even-money card game at the casino. The game will cost $10 to play. To play the game, players will draw one card at random If the card drawn is the Queen of Spades, they win bigtf the card is any other Queen, they get $15 (winning only $5). If the card is a Jack or an Ace, they get $20. If the card is any other spade, they ger $5 (ultimately losing $5). How much money should the players get if they draw the Queen of Spades? Remember, the goal is to make this an even- money game! 52 Queen of spudo s CAC in 516 ( 26

Explanation / Answer

Let P be the money one gets having drawn a queen of spades.

Let X be the random variable denoting the gain of a gambler.

So,

X = P- 10, when the queen of spades drawn; Prob( queen of spades) = 1/52

X = 15-10, when any other queen is drawn; Prob( any other queen) = 3/52

X = 20-10, when a jack or an ace is drawn; Prob( a jack or an ace) = 8/52

X = 5 - 10, when any other spade is drawn. Prob( any other spade) = 10/52 [ since a jack or an ace gives 20 and the queen gives P)

X = 0- 10. when any other card is drawn; Prob( any other card) = [52- (13 + 3 + 3 + 3)]/52 = 30/52.

Now, to make the game fair we must have the following condition.

E(X) = 0.

(P-10)*(1/52) + (15-10)*(3/52) + (20-10)*8/52 + (5-10)*10/52 + (0-10)*30/52 = 0

=> P + 3*15 + 20*8 + 5*10 = 10*52

=> P = 520 - 45 - 160 - 50 = 165

So, money on queen of spades should be $165.

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