King for a week
A certain tropical island has a ruling system consiting of twelve counsel members. Each week one of the twelve members of the counsel is picked to be king; at the end of the week a new king is picked at random from the twelve member counsel ( Note that it is possible for the same person to be elected king again, since the choice is completely random among the twelve). Question: how many weeks would elapse, on average, before each person has been king at least once? A more difficult question would be how many weeks would have elapsed before there was at least a 50% chance that everyone has been king at least once? The first question is asking for the mean, the second for the median.
Answer for the expected number of weeks is:
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