A fair 6-sided die is rolled repeatedly. (a) Find the expectednumber 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 consecutive1’s. (c) Let an be the expected number of rolls needed to get thesame value n times in a row (i.e., to obtain a streak of nconsecutive j’s for some not-specified-in-advance value of j). Finda recursive formula for an+1 in terms of an. Hint: Divide the timeuntil there are n+1 consecutive appearances of the same value intotwo pieces: the time until there are n consecutive appearances, andthe rest. (d) Find a simple, explicit formula for an for all n ? 1.What is a7 (numerically)?
