Q.7.3 - A First Course in Probability

Question

Twenty individuals consisting of $10$ married couples are to be seated at $5$ different tables, with $4$ people at each table.

 (a) If the seating is done“at random,” what is the expected number of married couples that are seated at the same table? 

(b) If $2$ men and $2$ women are randomly chosen to be seated at each table, what is the expected number of married couples that are seated at the same table?

Step-by-Step Solution

Answer

According to the information,

a)If the seating is done“at random,”  the expected number of married couples that are seated at the same table = $\frac{30}{19}$$\frac{30}{19}$

b) If $2$ men and $2$ women are randomly chosen to be seated at each table, the expected number of married couples that are seated at the same table is $2$

Step 1: Given Information (part a)

If the seating is done“at random,” what is the expected number of married couples that are seated at the same table 

Step 2: Explanation (part a)

Let X represents the number of married couples that are seated at the same table, and let's define indicator variables Ij as:

$I_j=\{\begin{array}{ll}1,& \text{if}{E}_j\text{occurs}\\ 0,& \text{if}{E}_j\text{does not occur}\end{array}$

Whereby $E_{j,} j=1,2,....,10,$denote the event

$E_{j }=$"j the 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,

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

Step 3: Explanation (part a)

Consider the next events:

${W_j}^i =$" Woman from j th married couple is at i th table",

${M_j}^i =$" Man from j th married couple is at i th table".

 Assume that the seating is done at random. Since there are 5 different tables, with 4 seats at each table, we have: 

$P\left\{E_j\right\}=P\{"j\text{th married couple is at 1st table}"\}$

$+\cdots +P\{"j\text{th married couple is at}5\text{th table}"\}=$

$P\left\{W_j^1\right\}P\{M_j^1\mid W_j^1\}+P\left\{W_j^2\right\}P\{M_j^2\mid W_j^2\}+\cdots +P\left\{W_j^5\right\}P\{M_j^5\mid W_j^5\}=$

$\frac{4}{20}\left(\frac{3}{19}\right)+\frac{4}{20}\left(\frac{3}{19}\right)+\cdots +\frac{4}{20}\left(\frac{3}{19}\right)=\frac{3}{19}$

$\text{According to}(\ast )\text{we get:}$

$E\left[X\right]=10\left(\frac{3}{19}\right)=\frac{\mathbf{30}}{\mathbf{19}}$

Step 4: Final Answer (part a)

If the seating is done“at random,”  the expected number of married couples that are seated at the same table  = $\frac{30}{19}$

Step 5: Given Information (part b)

If $2$ men and $2$ women are randomly chosen to be seated at each table, what is the expected number of married couples that are seated at the same table

Step 6: Explanation (part b)

Now, assume that $2$ men and $2$ women are randomly chosen to be seated at each table. We will consider $2$ men and $2$ women as two different objects. Therefore, with this consideration, instead of $20$ spots, we now have $10$ spots at $5$ different tables, $2$ spots at each table. Therefore, since we have $5$ 'pairs' of $2$ men and $5$ 'pairs' of $2$ women, we get:

$P\left\{E_j\right\}=P\{"j\text{th married couple is at 1st table}"\}$$+\cdots +P\{"j\text{th married couple is at}5\text{th table}"\}=$

$P\left\{W_j^1\right\}P\{M_j^1\mid W_j^1\}+P\left\{W_j^2\right\}P\{M_j^2\mid W_j^2\}+\cdots +P\left\{W_j^5\right\}P\{M_j^5\mid W_j^5\}=$

$\frac{2}{10}\left(\frac{1}{5}\right)+\frac{2}{10}\left(\frac{1}{5}\right)+\cdots +\frac{2}{10}\left(\frac{1}{5}\right)=\frac{1}{5}$

$\text{According to}(\ast )\text{we obtain:}$

$E\left[X\right]=10\left(\frac{1}{5}\right)=\mathbf{2}$

Step 7: Final Answer (part b)

If $2$ men and $2$ women are randomly chosen to be seated at each table, the expected number of married couples that are seated at the same table is $2$