### Solution Found!

# N people arrive separately to a professional dinner.Upon

**Chapter 7, Problem 7.8**

(choose chapter or problem)

**QUESTION:**

N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the \(\left(\begin{array}{l}

N \\

2

\end{array}\right)\) pairs of people is, independently, a pair of friends with probability p, find the expected number of occupied tables. Hint: Let \(X_{i}\) equal 1 or 0, depending on whether the ith arrival sits at a previously unoccupied table.

### Questions & Answers

**QUESTION:**

N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the \(\left(\begin{array}{l}

N \\

2

\end{array}\right)\) pairs of people is, independently, a pair of friends with probability p, find the expected number of occupied tables. Hint: Let \(X_{i}\) equal 1 or 0, depending on whether the ith arrival sits at a previously unoccupied table.

**ANSWER:**

Step 1 of 3

From the information, observe that there are N people arriving separately for a professional dinner.

Each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of \(\left( \begin{array}{l}N\\2\end{array} \right)\) pairs of people is, independently, a pair of friends with probability p.

Calculate the expected number of occupied tables.