We are tasked with finding the expected number of pairs of adjacent cards that are both hearts in a randomly arranged circular deck of 52 cards.

Step 1: Understanding the Problem

A standard deck contains 52 cards, divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit, including hearts, has 13 cards. The cards are arranged in a circle, so the adjacency between the last and first cards must also be considered.

Step 2: Defining Pairs of Adjacent Cards

In a circle of 52 cards, there are 52 adjacent pairs. Specifically, each card is adjacent to two others. Therefore, we can number the cards $C_1, C_2, \dots, C_{52}$, and the adjacent pairs are:

• $(C_1, C_2), (C_2, C_3), \dots, (C_{51}, C_{52})$
• Plus the pair $(C_{52}, C_1)$, since the arrangement is circular.

Step 3: Indicator Variables

Define an indicator variable $X_i$ for each adjacent pair, where:

\begin{cases} 1 & \text{if the pair is two hearts} \\ 0 & \text{otherwise} \end{cases}$$ Our goal is to find the expected value of the total number of adjacent heart pairs: $$X = \sum_{i=1}^{52} X_i$$ Thus, the expected number of adjacent heart pairs is: \[ \mathbb{E}[X] = \sum_{i=1}^{52} \mathbb{E}[X_i] \] ### Step 4: Computing $$\mathbb{E}[X_i]$$ Each $$X_i$$ represents the event that both cards in the $$i$$-th pair are hearts. Since there are 13 hearts in the deck, the probability that a randomly chosen card is a heart is: $$P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$$ The probability that both cards in an adjacent pair are hearts is: $$P(\text{both hearts}) = \frac{13}{52} \times \frac{12}{51} = \frac{13 \times 12}{52 \times 51} = \frac{156}{2652} = \frac{1}{17}$$ Thus, for each pair $$i$$, $$\mathbb{E}[X_i] = \frac{1}{17}$$ ### Step 5: Total Expected Number of Adjacent Heart Pairs Since there are 52 adjacent pairs, the expected number of adjacent heart pairs is: $$\mathbb{E}[X] = 52 \times \frac{1}{17} = \frac{52}{17} \approx 3.06$$ Thus, the expected number of adjacent pairs of cards that are both hearts is $$\frac{52}{17}$$, or approximately $$3.06$$. ### Final Answer: The expected number of adjacent pairs of hearts is $$\boxed{\frac{52}{17}}$$. --- Would you like further details on the steps or any clarifications? Here are some related questions for further exploration: 1. What is the probability of getting at least one adjacent pair of hearts? 2. How does the expected number of adjacent heart pairs change for a deck with more or fewer cards? 3. How would the calculation differ if the arrangement were linear rather than circular? 4. What is the variance in the number of adjacent heart pairs? 5. Can we generalize this approach to find the expected number of adjacent pairs for any suit? **Tip:** The expected value is a weighted average over all possible outcomes, so understanding the role of probability in each pair is crucial when calculating it for adjacent cards.