# Expected Number of rolls of 1 before first 6 is rolled

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.

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 $$5$$.