You roll a die until you get a 6. What is the expected sum of all rolls?

You roll a die until you get a 6. What is the expected sum of all rolls?

Approach: Use Wald's identity or conditioning on first roll.

21. Let N be number of rolls, X_i roll results. E[∑X_i] = E[N]E[X_i|game continues]. But careful: last roll is 6. Using conditioning: E[sum] = E[N] * 3.5 - 0.5 = 6*3.5 - 0.5 = 21.

Follow-up: What if you stop when sum exceeds 100?

Key concepts: Wald's identity, conditional expectation, geometric distribution.