Sanderson M. Smith
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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.
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