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.

Short answer

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

Step-by-step solution

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.

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.

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 $E=\sum _{i=1}^N(1-p{)}^{i-1}$.

Final answer 

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