Question

Suppose in Self-Test Problem $7.3$that the $20$people are to be seated at seven tables, three of which have $4$seats and four of which have $2$seats. If the people are randomly seated, find the expected value of the number of married couples that are seated at the same table.

Short answer

The expected value of the number of married couples that are seated at the same table is$\frac{22}{19}$.

Step-by-step solution

Given Information

$20$people are to be seated at seven tables, three of which have $4$seats and four of which have $2$seats.

Explanation

Indicator variables as,

whereby,

$E_j,j=1,2,\dots ,10$denotes the events.

$E\_j="j$th married couple is at the same table.

Then, $X=\sum _{j=1}^{10}{I}_j$.

and therefore the expected number of married couples that are seated at the same table is

$E\left[X\right]=E\left[\sum _{j=1}^{10}{I}_j\right]=\sum _{j=1}^{10}E\left[I_j\right]=\sum _{j=1}^{10}P\left\{E_j\right\}·(*)$

Explanation

Consider the next events,

$W_j^i="$woman from $j$th married couple is at$i$ th table " ,