Solution
A Multiple of Seven
Show the problem again
A fair six-sided die is rolled repeatedly and a running total of all rolls is kept. You stop as soon as the running total is a positive multiple of 7. What is the expected number of rolls? Give an exact number.
Worked solution
The answer is 7. The first roll gives a total between 1 and 6, which is never a multiple of 7. After that, suppose the running total is congruent to r mod 7 with r ≠ 0. The six faces 1 through 6 move the total to the six residues r + 1, ..., r + 6, which are all residues except r itself. Exactly one of these is 0, so every subsequent roll ends the game with probability exactly 1/6, regardless of history. Hence the number of rolls after the first is geometric with mean 6, and the total expected number of rolls is 1 + 6 = 7. No Markov chain needs to be solved; the uniformity across residues does all the work.
Source: Original problem written for AxiomIQ.