A fair 6-sided die is rolled repeatedly.

(a) Find the expected number of rolls needed to get a 1 followed right away by a 2. Hint: Start by conditioning on whether or not the first roll is a 1.

(b) Find the expected number of rolls needed to get two consecutive 1s.

(c) Let an be the expected number of rolls needed to get the same value n times in a row (i.e., to obtain a streak of n consecutive js for some not-specified-in-advance value of j).Find a recursive formula for an+1 in terms of an. Hint: Divide the time until there are n+1 consecutive appearances of the same valueinto two pieces: the time until there are n consecutive appearances, and the rest.

(d) Find a simple, explicit formula for an for all n 1. What is a7 (numerically)?