Sanderson M. Smith
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SOPHISTICATED PROBABILITY WITH SIMPLE DICE GAME
Here's the game description...
Players stand around a table.
Two dice are rolled.
Each player gets the total on the first roll as his/her score.
(*) A player who wants to keep that score sits down.
For those standing, the dice are rolled again.
If the total is 2, 3, or 12, all standing players get a game score of 0. They sit down.
If the total is 4,5,6,7,8,9,10, or 11, this is added to the scores of standing players.
Now continue by going back to (*). The game ends when all players are seated.
As a class activity, have the group play the game a determined number of times. The overall winner is the player who has the greatest total score after all of the games have been played.
Here are some questions relating to expectations for a single game.
What is a player's expected score if he/she
(1) sits down after the first roll?
(2) sits down after the second roll?
(3) sits down after the third roll?
(4) sits down after the n-th roll?
(5) Is there a best strategy to use in this game?
My responses and my reasoning follow:
If a player rolls two dice, the expected total is 7. If two dice are rolled twice, the expected total of the outcomes is 14. If two dice are rolled three times, the expected total of the outcomes is 21, etc.
(1) Response: If a player sits down after the first roll, the expected score is 7.
(2) Response: On the second roll, the probability that you "zero out" is 4/36. (You can get a "2" one way, a "12" one way, and a "3" two ways. The probability that you don't "zero out" is 32/36. If one sits down after the second roll, the expected score is (4/36)(0) + (32/36)(14) = 12.444.
(3) To get a positive total if one sits down after the third roll, then you must not "zero out" on the second and third rolls. The expected score is (21)(32/36)2 + (0)[1-(32/36)2] = 21(32/36)2 = 16.593.
(4) Following the pattern above, if a player sits down after the n-th roll, the expected score is (7n)(32/36)(n-1).
Now, if you examine the function y = (7x)(32/36)(x-1) you will see a curve that does have a maximum value at (8.49,24.6).
The following table shows expected scores if you sit down after n rolls.
n = Expected score 1 7 2 12.444 3 16.593 4 19.665 5 21.85 6 23.307 7 24.17 8 24.554 9 24.554 10 24.251 11 23.712 12 22.993 13 22.142 14 21.195 15 20.186 30 6.337 40 2.8328 50 1.0904 60 0.40295 70 0.14477 80 0.05095 90 0.01765
(5) By my reasoning, if you are going to play this game many times, you have the best chance of maximizing your total game scores by sitting down after the 8th roll.
After 10 games, if you always sit down after the 8th roll, your expected total score would be 10(24.554) = 245.54, or about 246 points.
After 10 games, if you always sit down after the 3rd roll, your expected total score would be 10(16.593) = 165.93, or about 166 points.
"We figured the odds as best we could, and then we rolled the dice."
President Jimmy Carter, June 10, 1976
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