Rolling a 2 or a 1, 1

Suppose you roll a fair six-sided die until you either roll a 2 for the first time or get two consecutive 1s. What is the expected number of rolls you will perform?

Answer

Let $$E$$ be the expected number of rolls starting with no prior 1, and $$E_1$$ be the expected additional rolls given the previous roll was a 1. Then, by conditioning on the outcome of the next roll, we have:

E = \frac{1}{6}\cdot1 + \frac{1}{6}\bigl(1+E_1\bigr) + \frac{2}{3}\bigl(1+E\bigr),

E_1 = \frac{1}{3}\cdot1 + \frac{2}{3}\bigl(1+E\bigr).

Solving these equations yields $E = \frac{21}{4}$ .