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Say we have a fair six-sided die, and we want to find the expected number of 1s rolled before the first six is rolled.

Apparently this can be solved by conditioning/recursion. I know that by that same method there are 5 expected rolls before rolling a 6 (with roll 6 being 6). Thus the expected number of 1s before that should be much less than 6.

Would I rewrite this problem to the expected number of 1s rolled in $6-1=5$ rolls (the rolls before the first six)? I'm not sure how to start solving this.

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If you know the number of expected rolls before first rolling a $6$, then this is the expected number of $1$s before the first $6$ plus the expected number of $2$s before the first $6$ plus ... plus the expected number of $5$s before the first $6$. But each of these is equal, by symmetry, so you just need to divide by $5$.

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