Sanderson M. Smith

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

(

_{3}C_{0})(1/6)^{0}(5/6)^{3}= 125/216 = .5787Probability (exactly one 3)

(

_{3}C_{1})(1/6)^{1}(5/6)^{2}= 75/216 = .3472Probability (exactly two 3's)

(

_{3}C_{2})(1/6)^{2}(5/6)^{1}= 15/216 = .0694Probability (three 3's)

(

_{3}C_{3})(1/6)^{3}(5/6)^{0}= 1/216 = .0046For 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, p

_{0}, p_{1}, p_{2}, and p_{3}, then the expected payout, g, for a player isE(g) = (p

_{0})(.5787) + (p_{1})(.3472) + (p_{2})(.0694) + (p_{3})(.0046)Here is a table demonstrating a few arbitrarily-chosen game costs, payouts, and expected player gain for Three-Dice Game.

Cost to play

p _{0}p _{1}p _{2}p _{3}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.

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