The Dice Game

[kirman]

Hello everyone,

 

I want to ask questions about my task. Below are the description. Thank you in advance.

 

Kind regards,

Kirman.

 

 

The aim of this task is to create a dice game in a casino and model this using

probability. It is important to examine how best to run the game from both the

perspective of a player and the casino. In doing so analyse the game to consider the

optimal payments by the player and payouts by the casino.

1. 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 Bob’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?

2. 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?

3. 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. 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.)

 

4. 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.

[CraigD]

Hi kirman – welcome to hypography! :)

 

Your homework questions are nicely written – with a bit of thought, you should be able to figure our their answers, and get a good intuitive introduction of systematic counting and probability.

 

As I guess you wouldn’t be looking to the internet and our humble forum for help if you weren’t at least slightly stuck, here’s a hint on problem #1:

 

Try drawing a 6 x 6 table, with rows for Ann’s red die and columns for Bob’s white, or the other way around:

  1  2  3  4  5  6
1  .  .  .  .  .  .
2  .  .  .  .  .  .
3  .  .  .  .  .  .
4  .  .  .  .  .  .
5  .  .  .  .  .  .
6  .  .  .  .  .  .

Now mark the cells where Ann wins, and count them. Divide this by the total number of cells (6x6 = 36), and you have Ann’s probability of winning.

 

Drawing and counting cells in tables gets more complicated for the later questions, but you should be able to find some ways to find the counts without actually drawing and counting. Once you have, you’ll have figured out how to solve problems like this in general.

 

If you learn to do this quickly in your head, you’ll also have the makings of a serious backgammon hustler or dice-based game designer, making these lessons among the rare ones that can be immediately helpful to making a living. :)