7.11 - A First Course in Probability

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.

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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

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

Step 3: Explanation

Consider the next events,

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