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A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls.

Consider \(n\) independent flips of a coin having probability \(p\) of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if \(n=5\) and the outcome is \(H H T H T,\) then there are 3 changeovers. Find the expected number of changeovers.

A group of \(n\) men and \(n\) women is lined up at random. (a) Find the expected number of men who have a woman next to them. (b) Repeat part (a), but now assuming that the group is randomly seated at a round table.

A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.

Cards from an ordinary deck of 52 playing cards are turned face up one at a time. If the 1 st card is an ace, or the 2nd a deuce, or the 3 rd a three, or \(\ldots\), or the 13 th a king, or the 14 an ace, and so on, we say that a match occurs. Note that we do not require that the \((13 n+1)\) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

How many times would you expect to roll a fair die before all 6 sides appeared at least once?

If \(X_{1}, X_{2}, \ldots, X_{n}\) are independent and identically distributed random variables having uniform distributions over \((0,1),\) find (a) \(E\left[\max \left(X_{1}, \ldots, X_{n}\right)\right]\) (b) \(E\left[\min \left(X_{1}, \ldots, X_{n}\right)\right]\)

If 101 items are distributed among 10 boxes, then at least one of the boxes must contain more than 10 items. Use the probabilistic method to prove this result.

There are 4 different types of coupons, the first 2 of which comprise one group and the second 2 another group. Each new coupon obtained is type \(i\) with probability \(p_{i}\) where \(p_{1}=p_{2}=1 / 8, p_{3}=p_{4}=3 / 8 .\) Find the expected number of coupons that one must obtain to have at least one of (a) all 4 types; (b) all the types of the first group; (c) all the types of the second group; (d) all the types of either group.

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain (a) 2 aces; (b) 5 spades; (c) all 13 hearts.