Solution
Roll Until Six
Show the problem again
You roll a fair die repeatedly until you roll a six, then stop. What is the expected sum of all rolls, including the final six?
Worked solution
The answer is 21. The number of rolls N is geometric with mean 6, and each roll has expectation 3.5 independent of when you stop (the stopping rule depends only on past rolls). By Wald's identity, E[sum] = E[N] · E[single roll] = 6 × 3.5 = 21.
Source: Standard geometric-distribution exercise from introductory probability texts. Statement written for AxiomIQ.