Answer to April 30, 2001 Problem

The Apple Blossom Problem
The Apple Blossom Festival is celebrated during the first weekend in May in Winchester. One enterprising Sherando student has set up the following dice roulette game at this year's festival. He hopes to earn his college spending money for next year.

The board is marked with squares numbered 1 through 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.

An enterprising Handley student decides to play the game. On which number or numbers should she bet? And in the long run, should she expect to win or lose money (in other words, what is the expected payoff?)?


Solution:

The Handley student should bet on #6 or #12, and she would expect to win $1.11 for each dollar spent (for a net gain of 11 cents on each dollar).

Examine the following multiplication table:

    1 2 3 4 5 6
1 1 2 3 4 5 6
2 2 4 6 8 10 12
3 3 6 9 12 15 18
4 4 8 12 16 20 24
5 5 10 15 20 25 30
6 6 12 18 24 30 36

From the table, you can see that the number 6 and the number 12 can each be produced in four different ways. Since these are the numbers which occur the most, you should bet on one of them.

If you place a bet, your chances of winning are 4/36 = 1/9.
Since you win $10 for each $1 bet, your expected return on a bet is 1/9 x $10 = $1.11.

So the Sherando student should cancel the game and hit the books! And the Handley student would expect to WIN in the long run.


Correctly solved by:

1. Evelyne Stalzer New Jersey
2. Richard K. Johnson La Jolla, California
3. Chip Schweikarth * Winchester, Virginia
4. George Gaither Winchester, Virginia
5. Tom Kelley Winchester, Virginia
6. Keith Mealy Cincinnati, Ohio
7. Bob Hearn Winchester, Virginia
8. John C. Funk Ventura, California
9. Geoff Keith Santa Monica, California
10. Bill Hall Wellington, Florida
11. John Lybarger Calhoun Community College, Alabama
12. Kirstine Wynn Winchester, Virginia
13. Joe Heintz Manchester, Tennessee
14. Renata Sommerville Austin, Texas
15. David Dixon Bennettsville, South Carolina
16. Rich Murray Ridgetown, Ontario
* Wrote a calculator program to solve the problem