We are tasked with finding the expected number of people who are shorter than both of their immediate neighbors, when $n$ people of distinct heights randomly take their seats around a circular table with $n$ chairs.

Problem Breakdown:

1. Since the table is circular, each person has two immediate neighbors, one on the left and one on the right.
2. We are interested in how many people are shorter than both of their neighbors, i.e., for a person in seat $i$, their height $h_i$ should satisfy: $h_i < h_{i-1} \quad \text{and} \quad h_i < h_{i+1}$ where $h_{i-1}$ and $h_{i+1}$ are the heights of their immediate neighbors.

Expected Value Calculation:

Let’s define an indicator random variable $X_i$ for each person in seat $i$:

\begin{cases} 1 & \text{if person in seat } i \text{ is shorter than both of their neighbors,} \\ 0 & \text{otherwise.} \end{cases}$$ We are interested in finding the expected value of the total number of such people, which is the sum of all $$X_i$$'s: $$\text{Total } = X_1 + X_2 + \dots + X_n$$ By linearity of expectation, the expected total number of such people is: \[ \mathbb{E}[\text{Total}] = \mathbb{E}[X_1] + \mathbb{E}[X_2] + \dots + \mathbb{E}[X_n] \] #### What is $$\mathbb{E}[X_i]$$? The probability that a given person is shorter than both of their neighbors depends only on the relative order of the heights of the three people involved (the person and their two neighbors). Since the heights are randomly assigned, there are 3! = 6 possible ways to arrange the three people. Out of these 6 possible arrangements of the three people, there are exactly 2 arrangements where the person in the middle is shorter than both of their neighbors (i.e., their height is the smallest of the three): - Person in the middle is the smallest, neighbors are taller. Thus, the probability that person $$i$$ is shorter than both of their neighbors is: $$\mathbb{P}(X_i = 1) = \frac{2}{6} = \frac{1}{3}$$ ### Final Calculation: Since this probability is the same for all people seated at the table, the expected number of people who are shorter than both of their immediate neighbors is: $$\mathbb{E}[\text{Total}] = n \times \frac{1}{3}$$ Thus, the expected number of people who are shorter than both of their neighbors is: $$\boxed{\frac{n}{3}}$$ ### Conclusion: For $$n$$ people randomly seated around a circular table, the expected number of people who are shorter than both of their immediate neighbors is $$\frac{n}{3}$$. --- Would you like further details or have any questions about the explanation? Here are 5 related questions to consider: 1. How would the problem change if the table were linear instead of circular? 2. What is the probability that no one is shorter than both of their neighbors? 3. What is the expected number of people who are taller than both of their immediate neighbors? 4. How would the problem change if the neighbors were not seated next to each other but were chosen randomly from the group? 5. Can we generalize this problem to different criteria, such as being shorter than only one neighbor? **Tip:** In problems involving symmetry and randomness, breaking things down with indicator variables and using the linearity of expectation often simplifies complex counting problems.