A fair 6-sided die is rolled repeatedly. (a) Find the expected number of rolls needed to...

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 1’s. (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 j’s 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 value into 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)?

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orchestra answered 10 months ago

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