# Investigate the probability of someone rolling a die and the probability of it landing on particular number for a player to win the game

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Introduction

We have been asked to investigate the probability of someone rolling a die and the probability of it landing on particular number for a player to win the game. For A to win he/she must roll a 1 and if he/she does this they have won the game. For B to win, first of all A must lose and they must roll 2 or a 3 and then they have won the game. For C to win they must roll a 4,5 or 6 and of course B must have lost. I have to investigate these tasks: 1. The probability of A, B or C winning. 2. Who will be the most likely winner? 3. Most likely length of the game. I have first of all drawn a tree diagram so it is easier to interpret and it is easier to see things visually: From this I tried to find the probability that no one wins in Round 1 and this is how I did it: P (LLL) = 1- (5 x 2 x 1) 6 3 2 P (LLL) = 1 - 5 18 P (LLL) ...read more.

Middle

1 x 5 6 18 = 0.00357 (The answer from the formulae) = 125 (The correct answer) 34992 I now divide 125 and I get the same answer. We now know that the formulae works for 34992 the fourth round but now I am going to test it for the third round. 1 x 5 6 18 = 0.0129 (Formulae answer) = 25 = 0.0129 1944 Again my formulae works but I will give it one more test. The chance of A winning in the tenth round: 1 x 5 6 18 = 0.00000164 (Formulae answer) = 0.00000164 (Actual answer) I have now tested my formulae 3 times and each time has come out with the correct answer. I have now come to the conclusion that this is correct. From this I think that I might have found a formula for B: 5 x 5 18 18 Again n is the round number and this is the formula for B winning in a particular round. I will now test my formulae to find the probability of winning in round 4: 5 x 5 18 18 = 0.00595 Actual answer: 0.00595 P (B) ...read more.

Conclusion

= 5 13 C = the same as B C = 5 13 I will now check this: 3 + 5 + 5 = 1 13 13 13 I know now that this is correct as they all add up to 1. From this we can see that B and C are the most likely winners in any round. In question 3 we must decide the most likely number of rolls of the dice before someone wins. This is how I worked it out: %Chance that A will win 1st round = 16.6 %Chance that B will win in 1st round = 27.7 I found that 16.6% + 27.7% is 44.4%. Therefore it is quite unlikely that you will have someone who has won as it is less than 50%. So, I added C: %Chance that C will win in 1st round = 27.7 This time it is adds up to be more than 50% (72.2) so you have a very good chance that it will be 3 rolls. From this I can also work out that there is a 27.7% chance that you will have no winner. So, the mostly likely number of rolls is 3. By James Vandenbussche 10-3 Mrs Hammond ...read more.

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