91Ó°ÊÓ

Define the indicator variables \(X_i\) as follows: - \(X_i = 1\) if there is a match at position \(i\), - \(X_i = 0\) otherwise Our goal is to find the expected value of the total number of matches, which is equal to \(\mathrm{E}(\sum_{i=1}^{52} X_i)\).

By the linearity of expectation, we have: \(\mathrm{E}(\sum_{i=1}^{52} X_i) = \sum_{i=1}^{52} \mathrm{E}(X_i)\) We will now compute \(\mathrm{E}(X_i)\) for each \(i\) in the deck.

To compute \(\mathrm{E}(X_i)\), we will use the fact that the expected value of an indicator variable is equal to the probability that the event occurs: \(\mathrm{E}(X_i) = \mathrm{P}(X_i = 1)\) To find \(\mathrm{P}(X_i = 1)\), we need to find the probability of a match occurring at position \(i\). Notice that for every 13 cards, there is exactly one card of each rank, so the probability of a match at position \(i\) is the same for all positions that have the same remainder when divided by 13. In general, there are four cards of each rank and each rank will appear four times within the sequence of remainders. Therefore, for any \(i\) such that \(1\leq i\leq52\) and regardless of the value of \(i \pmod{13}\), we have: \(\mathrm{P}(X_i = 1) = \dfrac{4}{52}= \dfrac{1}{13}\)

Now we can compute the expected number of matches by summing over all positions in the deck: \(\mathrm{E}(\sum_{i=1}^{52} X_i)=\sum_{i=1}^{52} \mathrm{E}(X_i)=\sum_{i=1}^{52} \mathrm{P}(X_i = 1)=\sum_{i=1}^{52} \dfrac{1}{13} = \dfrac{52}{13}=4\) So the expected number of matches that occur in the deck is 4.