Chuck-a-Luck

Rules:

· Player plays against “house”.

· Player bet’s any amount on a number 1, 2, 3, 4, 5, or 6. For simplicity let’s say $1.

· Then three dice are rolled simultaneously.

· If player’s number comes up on one die, player wins $1

· If player’s number comes up on two dice player wins $2

· If player’s number comes up on three dice, player wins $3

· If player’s number does not come up on any of the three dice, player loses the $1 that was bet.

Superficial analysis (WRONG):

Three dice are rolled so tne chance of the player’s number coming up is 3/6 or ½. So on a typical roll player breaks even (wins ½ and loses ½). But every so often player wins $2 or $3 to show a profit.

Correct analysis:

The probability that the roll contains none of the player’s number is :

125/216 = .5787…

T he probability that the roll contains exactly one of the player’s number is :

75/216= .3472…

T he probability that the roll contains exactly two of the player’s number is :

15/216=.0694…

T he probability that the roll contains exactly three of the player’s number is :

1/216=.0046…

Thus the expected value to the player of a $1 bet is:

(125/216)·(-1)+(75/216)·1+(15/216)·2+(1/216)·3 = -0.078

That is the player will lose about 8 cents on the average.