# Dice Game: Creating Probabilistic Models for a Casino

*The Dice Game: Creating Probabilistic models *

a. Consider a game with two players, Ann and Bob. Ann has a red die and Bob a white die. They roll their dice and note the number on the upper face. Ann wins if her score is higher than Bobâ€™s (note that Bob wins if the scores are the same). If both players roll their dice once each what is the probability that Ann will win the game?

b. Now consider the same game where Ann can roll her die a second time and will note the higher score of the two rolls but Bob rolls only once. In this case what is the probability that Ann will win?

c.Investigate the game when both players can roll their dice twice, and also when both players can roll their dice more than twice, but not necessarily the same number of times.

d.Consider the game in a casino where the player has a red die and the bank has a white die. Find a model for a game so that the casino makes a reasonable profit in the case where the player rolls the red die once and the bank rolls the white die once. (When creating your model you will need to consider how much a player must pay to play a game and how much the bank will pay out if the player wins. Do this from the perspective of both the player and the casino and consider carefully the criteria for whether the game can be considered worthwhile for both the player and the casino.)

e. Now consider other models for the game including cases such as where the player or the bank rolls their dice multiple times, or where multiple players are involved in the game.

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You might tell us what you have tried so far. –  Henry Aug 7 '12 at 8:10
for part a., i drew a probability grid for Ann and Bob and found the chances of Ann winning and the changes of Bob winning and the chances that neither win... they are as follows: P(A) = 15/36 P(B) = 6/36 P(N) = 15/36... What are some other ways of doing it ? .. –  Math_2nd_ED Aug 7 '12 at 8:36
For this one, can take advantage of symmetry. Let $x=\Pr(A\gt B)$. By symmetry, $\Pr(B \gt A)=x$. But $\Pr(A=B)=\frac{1}{6}$. So $2x+\frac{1}{6}=1$. It follows that $x=\frac{5}{12}$. We can also do it by a grid. Your calculation was along the right lines, but if $N$ then Bob wins. And things went wrong or there is a typo, your calculation should give $P(N)=\frac{6}{36}$. –  André Nicolas Aug 7 '12 at 10:00