A deck of n cards, numbered 1 through n, is thoroughly...

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 $i$ th 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 $$ \begin{aligned} E[N] &=\frac{1}{n}+\frac{1}{n-1}+\cdots+1 \\ & \approx \int_{1}^{n} \frac{1}{x} d x=\log n \end{aligned} $$ (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 that maximizes $E[N]$ is one which 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 $$ \begin{aligned} E[N] &=1+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !} \\ &=e-1 \end{aligned} $$

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 $i$ th 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
$$
\begin{aligned}
E[N] &=\frac{1}{n}+\frac{1}{n-1}+\cdots+1 \\
& \approx \int_{1}^{n} \frac{1}{x} d x=\log n
\end{aligned}
$$
(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 that maximizes $E[N]$ is one which 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
$$
\begin{aligned}
E[N] &=1+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !} \\
&=e-1
\end{aligned}
$$

Key Concepts

Series Approximations and Asymptotic Behavior

In the analysis of the expected number of correct guesses, series such as the harmonic series and truncated exponential series often appear. For example, the sum of reciprocals approximates the harmonic series, which, for large n, behaves logarithmically. Similarly, partial sums of reciprocal factorials approximate constants like e – 1. Recognizing these patterns and using integral approximations or limit behaviors is key in understanding and simplifying the asymptotic behavior of complex summations in probability.

Strategy Optimization in Probabilistic Settings

This concept deals with finding the strategy that maximizes the desired outcome—in this case, the number of correct guesses. In various feedback scenarios, such as when full card information is revealed versus simply being told if a guess is right or wrong, the optimal guessing strategy changes. Understanding how to modify tactics based on the available information is essential in decision theory and probability, and it is used to derive the maximum expected payoff for the different variants of the problem.

Sequential Decision-making with Feedback

This involves making a series of decisions where each decision is based on both prior actions and new information that becomes available. In the card guessing game, the feedback provided after each guess (whether the card is revealed, or only whether the guess is correct) enables the following guesses to be adjusted. This concept is crucial for understanding how strategies can be optimized depending on the type and amount of feedback received, thereby potentially increasing the overall expected number of correct guesses.

Linearity of Expectation

The principle of linearity of expectation states that the expected value of the sum of random variables is the sum of their individual expected values, regardless of whether they are independent. In the context of the card guessing problem, this allows us to calculate the total expected number of correct guesses by summing the individual probabilities of a correct guess at each position, resulting in neat expressions even when the guesses are made sequentially.

Uniform Random Permutations

This concept refers to the idea that when a deck of n cards is thoroughly shuffled, every possible ordering of the cards is equally likely. In problems involving random permutations, such as this one, the probability of any particular card being in any particular position is uniform, which leads to many elegant symmetry arguments and simplifies the calculation of expected values due to the equal likelihood of outcomes.

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