MHB What is the Probability of Winning a Dice Game?

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The discussion focuses on the probability calculations for a dice game from a past exam paper. The poster expresses uncertainty about their answers to several parts of Question 3, particularly regarding the probabilities of rolling specific sums. For parts (c,i) and (c,ii), there is confusion over the correct approach to calculating probabilities, with suggestions that the answers may differ from what was initially calculated. Additionally, the poster struggles with part (d), where they provide an answer but doubt its accuracy. Overall, the conversation emphasizes the complexities of probability in dice games and the need for clarification on specific questions.
pretzel1998
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Hi There!

This is my first time posting on this forums, so I hope I am following the right etiquette.

Im having some severe difficulties with the problems under Question 3 in this past exam paper. http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2014/91267-exm-2014.pdf
Can someone please check my answers and explain the questions where I faceplanted? Thanks

3. (a), I got 1/9 which I am pretty sure is good.
3. (b), I got 2/3 which I am pretty sure is good.
3. (c,i). I got 1/81, I am not very confident about this answer because the wording of the question is incredibly bad. I did 4/36 X 4/36 as that's the probability he gets 5 on the first roll, and then 5 on the second roll to win?
3. (c,ii). I got 13/1458, I don't think I got this right because it seems like such a crazy probability. I did the probability he got the sum of 5 on the first roll (4/36), multiplied by the probability that he did not get a sum of 5 on the second roll (13/18 I think? I might be wrong), the multiplied again by the probability that he will get a sum of 5. so in total (4/36 X 13/18 X 4/36 = 13/1458)? Really confused about this question.
3. (d) This question I had absolutely no idea what to do. I some how got 7/18, but I think I am wrong.

Thanks so much!
 
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pretzel1998 said:
Hi There!

This is my first time posting on this forums, so I hope I am following the right etiquette.

Im having some severe difficulties with the problems under Question 3 in this past exam paper. http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2014/91267-exm-2014.pdf
Can someone please check my answers and explain the questions where I faceplanted? Thanks

3. (a), I got 1/9 which I am pretty sure is good. I agree.
3. (b), I got 2/3 which I am pretty sure is good. I agree.
3. (c,i). I got 1/81, I am not very confident about this answer because the wording of the question is incredibly bad. I did 4/36 X 4/36 as that's the probability he gets 5 on the first roll, and then 5 on the second roll to win? I think that for part (c) you are meant to assume that he got a 5 on the first roll, so you only need to find the probability of scoring 5 on the second roll. If so, then the answer would be 1/9 rather than 1/81.
3. (c,ii). I got 13/1458, I don't think I got this right because it seems like such a crazy probability. I did the probability he got the sum of 5 on the first roll (4/36), multiplied by the probability that he did not get a sum of 5 on the second roll (13/18 I think? I might be wrong), the multiplied again by the probability that he will get a sum of 5. so in total (4/36 X 13/18 X 4/36 = 13/1458)? Really confused about this question. Here again, if you ignore the 1/9 that came from the first roll, you would get 13/162 rather than 13/1458, and I think that would be the answer they are looking for.
3. (d) This question I had absolutely no idea what to do. I some how got 7/18, but I think I am wrong.
On the first roll, M has a 2/9 probability of winning and W has a 1/9 probability. To make the game fair, the second roll must be arranged so that M's and W's overall probability of winning are both 1/2. The probability of a second roll being necessary is 2/3. If the second roll is arranged in such a way that M's probability of winning it is 15/36 (and W's probability of winning it is 21/36), then M's chances of winning the game are 2/9 (on the first roll) plus (2/3)x(15/36) (on the second roll). That comes to $\color{red}{\frac29 + \frac5{18} = \frac12}$, as required. So I think the answer should be 15/36.

Thanks so much!
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