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If I roll a 6-sided die x times (or roll x 6-sided dice at once), what is the probability that the sum of the result is greater than another y rolls (or one roll of y dice)?

Fabian Allendorf
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  • Parts of the solution are given at https://stats.stackexchange.com/questions/291549, https://stats.stackexchange.com/questions/392943, and https://stats.stackexchange.com/questions/3614. https://stats.stackexchange.com/a/116913/919 includes software to (easily and naturally) compute the answer. Once `x` and `y` grow large (around 10 or so), Normal approximations work well (provided some care is taken also to estimate the chance of a tie). – whuber Oct 09 '21 at 15:08
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    Note that for fair six-sided dice, if V is the outcome on one die, 7-V has the same probability distribution as V. So you can use this to get all the dice terms on one side of the inequality. Let $S_x$ and $T_y$ be the totals for the x and y dice respectively. Then $P(S_x>T_y)$ $=$ $P(S_x>7y-T_y)$ $=$ $P(S_x+T_y>7y)$. That is, it's the same as the probability that the total on $x+y$ dice exceeds $7y$. So for example, the chance that a sum on 3 dice exceeds a sum on 2 dice is the same as the sum of 5 dice exceeding 14. ... ctd – Glen_b Oct 09 '21 at 23:35
  • ... ctd ... Now the probabilities of sums on small numbers of dice are easy to generate (e.g. in a spreadsheet, such as described here: https://stats.stackexchange.com/questions/3614/how-to-easily-determine-the-results-distribution-for-multiple-dice/3625#3625). This is fairly quick and easy up to about 10 total dice or so. Beyond that a normal approximation will usually be adequate, but some of the methods whuber points to are better if you need good accuracy further into the tails. – Glen_b Oct 09 '21 at 23:37
  • Thank you that helped me very much. I guess using Troll software to determine the probability distribution of sum of dice is the best way to go. Very interesting that probability of x dice exceeds y dice is the same as x+y dice exceeds 7y, I would never have guessed that – Fabian Allendorf Oct 10 '21 at 11:27

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