You roll a fair six-sided die repeatedly and keep a running total. What is the probability that the running total is ever exactly 100?

You roll a fair six-sided die repeatedly and keep a running total. What is the probability that the running total is ever exactly 100?

Approach: Treat the visited totals as a renewal process. The long-run density of visited points is the reciprocal of the mean step, so check how fast the transient from the start decays before quoting that limit at 100.

2/7. Let u_n be the probability the running total ever equals n. The recursion u_n = (u_{n-1} + ... + u_{n-6})/6 with u_0 = 1 is a renewal recursion, and the renewal theorem says u_n tends to the reciprocal of the mean step size, 1/3.5 = 2/7. The transient decays geometrically because the remaining roots of the characteristic polynomial lie strictly inside the unit circle, and at n = 100 the recursion gives 0.285714 to six decimals, indistinguishable from the stationary density 2/7. So the total misses any given large number 5/7 of the time. A die with faces 1 to k would hit it with probability 2/(k+1).

Follow-up: What is the probability the running total hits both 100 and 107?

Key concepts: renewal theorem, recursion, mean step size, stationary density.