Q. 7.8 - A First Course in Probability

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 $i$th arrival sits at a previously unoccupied table. 

Step-by-Step Solution

Answer

The expected number of occupied tables value found to be $E=\sum _{i=1}^N(1-p{)}^{i-1}$.

Step 1: Given Information

$N$ - people arrive separately. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend.

Find the expected number of occupied tables. 

Step 2: Explanation

Each person looks to see if he or she has any friends among those present, and sits either at the table of a friend or at an unoccupied table if none of those present is a friend.

Each of the$\left(\begin{array}{l}N\\ 2\end{array}\right)$.

Pairs of people is, independently, a pair of friends with probability $p$.

Let $X_i=1$ if the $i^{th}$arrival sits at a previously unoccupied table and,

$X_i=0$ otherwise.

Step 3: Explanation

Let us find the expected number of occupied tables.

The total number of occupied tables after all$N$ people have arrived is$\sum _{i=1}^N{X}_i$

$E\left[\sum _{i=1}^N{X}_i\right]=\sum _{i=1}^{N}E\left(X_i\right)$

$E\left[X_i\right]=P\left(i^{th}\right.$arrival has no friends amongst previously arrived $i-1$ people$)$

$=(1-p{)}^{i-1}$.

Hence, the expected number of occupied tables is $E=\sum _{i=1}^N(1-p{)}^{i-1}$.

Step 4: Final answer

The expected number of occupied tables value found to be $E=\sum _{i=1}^N(1-p{)}^{i-1}$.