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Basically, if I roll a die $x$ times, what is the probability that the average of all the results will be within $p$ of the actual average value of the die?

For example, say I roll a six-sided die $6$ times. I want to know how likely that the average value of the results is within $3.5\pm p$, where $p=0.1$.

I think I've been able to figure out the probability of the average being EXACTLY the die's average.... Basically in order for it to be exact, half of the rolls need to be matched by a roll that balances it out. Like a 1 balances a 6, or a 3 balances a 4. So for 10 rolls, the probability of it being exactly the average would be $\left(\frac16\right)^5$.

I don't see where to go to make it CLOSE to the average, but not necessarily exactly the average.

saulspatz
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James Bearss
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  • Welcome to math SE. have a look at mathjax to typeset your mathematical expressions. – Alain Remillard Jan 22 '20 at 20:34
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    It may be helpful to think of this as a sum problem rather than a mean problem - when rolling N dice, if you want their average to be 3.5+/-p, their sum needs to be between 3.5N-pN and 3.5N+pN. The formula for the likelihood of hitting a particular sum can be found at https://math.stackexchange.com/questions/2462142/probability-that-the-sum-of-n-dice-is-at-least-x, but the formula is a bit ugly, so with sufficient dice you may prefer a normal approximation as seen at https://stats.stackexchange.com/questions/46872/how-to-compute-the-distribution-of-sums-when-rolling-n-dice-with-m-faces. – Nuclear Hoagie Jan 22 '20 at 20:44

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