Sanderson M. Smith

THREE-DICE GAME...A STUDENT-INVENTED CASINO GAME

Three-Dice Game was invented by Whitney Abbott and Teddy Lee, Cate School Class of 1990.

In this game, a player pays \$3 for the opportunity to roll three dice. The player wins \$5 if exactly one die shows 3, \$10 if exactly two of the dice show 3, and \$100 if all three dice show 3.

Here are probabilities related to this Three-Dice Game:

 Probability (no 3's) (3C0)(1/6)0(5/6)3 = 125/216 = .5787 Probability (exactly one 3) (3C1)(1/6)1(5/6)2 = 75/216 = .3472 Probability (exactly two 3's) (3C2)(1/6)2(5/6)1 = 15/216 = .0694 Probability (three 3's) (3C3)(1/6)3(5/6)0 = 1/216 = .0046

For the payouts established by Whitney and Teddy, the expected payout, g, for a player is

E(g) = (\$0)(.5787) + (\$5)(.3472) + (\$10)(.0694) + (\$100)(.0046) = \$2.89

Since the player has paid \$3 to play, the casino expects to gain \$0.11 for each game played. For each dollar "invested" in Three-Dice Game, the casino expects to earn \$0.11/3 = \$.0367.

If the payouts for 0,1,2,3 three's are, respectively, p0, p1, p2, and p3, then the expected payout, g, for a player is

E(g) = (p0)(.5787) + (p1)(.3472) + (p2)(.0694) + (p3)(.0046)

Here is a table demonstrating a few arbitrarily-chosen game costs, payouts, and expected player gain for Three-Dice Game.

 Cost to play p0 p1 p2 p3 Expected payout = E(g) Expected gain for player Expected gain per \$1 spent. Who comes out ahead in the long run? \$3 \$0 \$5 \$10 \$100 \$2.89 -\$0.11 -\$.0367 casino \$5 \$0 \$0 \$100 \$500 \$9.24 \$4.24 \$0.848 player \$6 \$0 \$6 \$12 \$500 \$5.22 -\$0.78 -\$0.130 casino \$10 \$0 \$10 \$20 \$1000 \$9.46 -\$0.54 -\$.0540 casino

Three-Dice Game can easily be played in the classroom if dice are available. The game can also be simulated on the TI-83. Simply enter randInt(1,6,3) on the home screen and hit the ENTER key to see the simulated results of rolling three dice. A repeated pressing of ENTER will allow you to roll the three dice multiple times.