Sanderson M. Smith

The game of SPADES was invented by Liza Miller, Cate School Class of 1990.

A player is dealt five cards from a randomly shuffled deck of 52 cards. The player is then paid \$1 for each spade in the dealt hand. What should it cost to play this game is the casino wants to make money on it?

Liza's original investigation set the cost to play at \$1. Let's examine this situation.

It is, of course, necessary to find the probability of being dealt x spades, where x is a random variable in the set {0,1,2,3,4,5}. The probability function is

P(x) = (13Cx)(39C(5-x))/(52C5)

 Value of x 0 1 2 3 4 5 Probability 0.22153 0.41142 0.27428 0.08154 0.01073 0.000495 Payout \$0 \$1 \$2 \$3 \$4 \$5

If g represents the expected payoff for a player, then the expected value of g is

E(g) = (0)(.22153) + (1)(.41142) + (2)(.27428) + (3)(.08154) + (4)(.01073)+(5)(.000495) = 1.25

So, for each \$1 spent to play SPADES, the player expects to receive \$1.25. This represents an expected gain of \$0.25 per game.

Obviously, considering the cost originally set by Liza to play SPADES and the expected payoffs, this game is a money loser for a casino. The casino would have to make adjustments to make this game profitable.

It is interesting to note at this point that since a player received \$1 for every spade, and the expected number of spades in a 5-card hand is 1.25, the player should expect to receive a payoff of \$1.25 per hand. Since the player paid \$1 to play, the expected gain is \$0.25 per game. If the casino charged \$1.50 to play the game and continued to pay out \$1 for each spade in a player's hand, then the casino would expect to gain \$0.25 per game.

The casino could make adjustments in the payouts. If the payouts for x = 0,1,2,3,4,5 are, respectively, p0, p1, p2, p3, p4, p5, then the expected payoff to a player is

E(g) = (p0)(.22153) + (p1)(.41142) + (p2)(.27428) + (p3)(.08154) + (p4)(.01073)+(p5)(.000495)

Here is a table demonstrating a few arbitrarily-chosen game costs, payouts, and expected player gain for the game of SPADES.

 Cost to play p0 p1 p2 p3 p4 p5 Expected payout= E(g) Expected gain for player Expected gain per \$1 spent Who comes out ahead in the long run? \$1 \$0 \$1 \$2 \$3 \$4 \$5 \$1.25 \$0.25 \$0.250 player \$2 \$0 \$1 \$2 \$3 \$4 \$5 \$1.25 -\$0.75 -\$0.375 casino \$5 \$0 \$0 \$5 \$20 \$100 \$1000 \$4.57 -\$0.43 -\$0.086 casino \$5 \$0 \$0 \$0 \$10 \$15 \$5000 \$3.45 -\$1.55 -\$0.310 casino \$10 \$0 \$0 \$20 \$30 \$40 \$50 \$8.39 -\$1.61 -\$0.161 casino \$10 \$0 \$10 \$20 \$30 \$40 \$50 \$12.50 \$2.50 \$0.250 player

Spades can be played in the classroom using poker chips for money. It does, of course, take time to thoroughly shuffle decks of cards, deal five cards, examine the number of spades, reshuffle, etc., etc.

It is possible to simulate dealing five cards from a shuffled deck using your TI-83. The cards can be represented by the integers 1,2,3,...,50,51,52 with the integers 1,2,3,...,11,12,13 representing spades. In what follows, -> represents STO (Store).

seq(x,x,1,52,1)->L1 will put the integers 1 through 52 in list L1

rand(52)->L2 will put fifty-two random numbers in list L2

SortA(L2,L1) will put the numbers in L2 in ascending order with the cells in L1 going along for the ride.

Check the list L1. The first five numbers represent the cards dealt to a player, and any integer between 1 and 13 inclusive represents a spade.

This process can be repeated many times... and done much more quickly than going through the process of shuffling cards.