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N people arrive separately to a professional dinner.Upon

Chapter 7, Problem 7.8

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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.

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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.

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