You roll a fair die repeatedly until you have seen both a 1 and a 6 at least once. What is the expected number of rolls?
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Let E0 be the expected rolls from the start (seen neither). Let E1 be the expected additional rolls after seeing exactly one of {1,6}. From E1: each roll has 61 chance of seeing the missing value, so E1=6. From E0: each roll has 62=31 chance of seeing a 1 or 6. So E0=1+32E0+31E1, giving 3E0=1+2=3, so E0=9.