The expected number of throws of $n$-faced dice till the sum of number is a multiple of $m$

Asked Oct 13 '17 at 20:04

Active Nov 03 '23 at 02:40

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Consider a fair dice with $n$ sides, numbered from $1$ to $n$. Imagine this dice is rolled over and over until the cumulative sum of the rolls is divisible by $m$. The question we seek to answer is: What is the average number of rolls required to reach this condition?

There is a theorem: For $6$-faced dice, the expected number of throws till the sum of the number is a multiple of $m$ is $m$.

What's the result for $n$-faced dice?

edited Nov 03 '23 at 02:40

asked Oct 13 '17 at 20:04

maplemaple

1,383

Perhaps this https://math.stackexchange.com/questions/104299/dice-problem-throws-necessary-for-sum-multiple-of-n?rq=1 could help? – Studentmath Oct 13 '17 at 20:50
@Studentmath What's the result of $n$-faced dice? – maplemaple Oct 13 '17 at 23:50
In the answer to the linked question, it states it does not matter how many sides the dice has - the argument still holds. – Studentmath Oct 14 '17 at 06:32

The expected number of throws of $n$-faced dice till the sum of number is a multiple of $m$

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