A deck of n cards numbered 1 through n is

A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.
(a) If you are not given any information about your earlier guesses show that, for any strategy, E[N] = 1.
(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that, under this strategy,
A deck of n cards numbered 1 through n is

(c) Suppose that you are told after each guess whether you are right or wrong. In this case, it can be shown that the strategy which maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that
E[N] = 1 + 1/2! + 1/3! + · · · + 1/n!
‰ˆ e ˆ’ 1
For all parts, express N as the sum of indicator (that is, Bernoulli) random variables.


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