# Dice Game

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Introduction

Dice Game Three players A, B and C play a game with a single dice. The rules of the game are: Player A ALWAYS goes first. A rolls the dice. If the dice lands showing a 1 then A wins the game. If A does not role a 1 then B takes a turn. B rolls the dice. If the dice lands showing a 2 or 3 then B wins the game. If B does not roll a 2 or 3 then C takes a turn. C rolls the dice. If the dice lands showing a 4, 5 or a 6 then C wins the game. If C does not roll a 4,5 or a 6 then A starts again. This procedure continues until there is a winner. Investigate any or all of: The probabilities of each of A, B or C winning the game. Who will be the most likely winner? 3. The most likely length of the game in terms of the number of rolls of the dice to produce a winner. Playing the game. Before we started to work out all the probabilities for the questions we played the game to give us an idea of what th outcome should roughly be. ...read more.

Middle

The answers are all multiples of the first one. Formula of A winning = 1/6 (5/6 x 4/6 x 3/6)^N-1 N = the number of rounds minus the first round where A did not win otherwise there would not be a probability there. This is the probability to calculate the overall probability of A winning in any round. For this formula to work you must substitute the N with the round number for which you are trying to calculate the probability. Formula of B or C winning = 5/18^(N) This is the formula for B and C winning in any round apart from the first. For this formula to work, substitute the N for the round that you wish to calculate the probability for. To find the overall probability of each player A, B or C winning the game you need to use a geometric series. This is a simple example of a geometric series: X = 32 + 16 + 8 + 4 + 2 + 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ...... In this line you can see that each number is half the previous number 1/2 X = 16 + 8 + 4 + 2 + 2 + 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ...... ...read more.

Conclusion

From looking at the overall probabilities of A, B and C winning the overall game you can see that A by far has the worse chance of winning the game. The probabilities of of B and C winning the game are identical so they both have the same chance of winning the game, so there is not 1 clear player who is most likely to win the game. At first I thought that player B was the most likely player to win the game because I thought the overall probabilities were in fact the probabilities of each player rolling one of their allocated numbers. but then I realised that this was wrong because B only has 2 numbers (2 and 3) and C has 3 numbers (4,5 and 6) so if what I thought was correct then C would have the better probability and they would not be the same. The probabilities are really the probability of the player rolling one of their numbers and the order that they roll in put together. Because although B only has 2 numbers it rolls before C so it has chance to win before C gets to have a go. This order increases the probability of B winning and decreases the probability of C winning the game overall. ...read more.

This student written piece of work is one of many that can be found in our GCSE Pythagorean Triples section.

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