September 2019
Problem of the Month

Dice Roulette
by Bob Stanton

Our local casino has introduced a new game called Dice Roulette.   The board is marked with squares numbered from 1 to 36, and players bet by placing chips on these numbers.

Then a player rolls a pair of standard six-sided dice, and the winning number is the product of the values on the dice.

For example, if the dice show 3 and 5, the winning number is 15.   Players who bet on the winning number win $10 for every $1 they wager; the others lose.

If you decide to play this game, on which numbers should you bet?   And in the long run, should you expect to win or lose money?

You must show your work; be specific.

Solution to the Problem:

You should bet on 6 or 12, each of which can be produced four different ways (6 will win on a throw of 1-6, 2-3, 3-2, or 6-1; 12 will win on 2-6, 3-4, 4-3, or 6-2).

If you place such a bet, your chance of winning is 4/36 or 1/9.   Since you win $10 for a successful bet of $1, your expected return on a $1 bet is 1/9 of $10 or about $1.11.   Therefore, you should win in the long run (and the casino should discontinue the game).

Correctly solved by:

1. James Alarie	Flint, Michigan
2. Briggin Bluemel	Mountain View High School, Mountain View, Wyoming
3. Kelly Stubblefield	Mobile, Alabama

Send any comments or questions to: David Pleacher