Q. 4.17 - A First Course in Probability

Question

A total of $2n$ people, consisting of $n$ married couples, are randomly divided into $n$ pairs. Arbitrarily number the women, and let $W_i$ denote the event that woman $i$ is paired with her husband.

1. Find $P\left(Wi\right).$
2. For $i\ne j,$ find $P\left(W_i\mid W_j\right)$.
3. When $n$ is large, approximate the probability that no wife is paired with her husband. 
4. If each pairing must consist of a man and a woman, what does the problem reduce to? 

Step-by-Step Solution

Answer

1. $P\left(W_i\right)=\frac{1}{2n-1}$
   
2. $P\left(W_i\mid W_j\right)=\frac{1}{2n-3}$
   
3. The probability is $e^{-\left(0.5\right)}$
4. If the pairing must consist of a men and women, then this experiment reduces to match problem.  

Step 1: Given Information (part a)

Total people $2n$ on that $n$are couples.

They are randomly divided into $n$ pairs.

Here, $W_i$ is the event and $i th$ woman paired with her husband.

Step 2: Explanation (Part a)

Number of ways to $i$ th woman can paired with others $=2n-1$

Number of ways to $i th$ woman can paired with her husband $=1$

woman $i$ is ready to pair with any of the remaining $"2n-1"$people.

So the probability to pair with her husband is $P\left(W_i\right)=\frac{1}{2n-1}$

Step 3: Final answer (part a)

$P\left(W_i\right)=\frac{1}{2n-1}$

Step 4: Given Information (Part b)

Total people $2n$on that $n$ are couples.

They are randomly divided into $n$ pairs.

Here, $W_i$ is the event.

Step 5: Explanation (Part b)

In this case, we will see the probability of $i$ th woman to pair with her husband.

But, $i$ is likely to be pair with any people.

That means, $2n-3$

Number of ways $i$ th woman to pair with her husband is :$1$

Hence, the probability will be $P\left(W_i\mid W_j\right)=\frac{1}{2n-3}$

Step 6 : Final Answer (Part b)

$P\left(W_i\mid W_j\right)=\frac{1}{2n-3}$

Step 7: Given Information (Part c)

Total people $2n$on that $n$ are couples.

They are randomly divided into $n$pairs.

Here, $W_i$ is the event.

Step 8 : Explanation (Part c)

For large sample size $n$

It becomes Poisson distribution with mean $\lambda$ equal to the probability that no couple is paired together.

If $n$ is large then the possibility to pair with their husbands will approximately be Poisson with the following mean

$\lambda =\sum _{i=1}^n P\left(w_i\right)$ 

$=\frac{n}{2n-1}$

$\lambda =\frac{1}{2}$

probability that no wife is paired with their husband is:

$p\left(w_i=0\right)=\frac{e^{-\lambda }{\lambda }^0}{0!}$

$=\frac{e^{-\left({\displaystyle \frac{1}{2}}\right)}{\left(\frac{1}{2}\right)}^0}{1}$

$=e^{-(1/2)}$

Therefore, the probability is $e^{-\left(0.5\right)}$

Step 9: Final Answer

The probability is $e^{-\left(0.5\right)}$

Step 10: Given Information (Part d)

Total people $2n$ on that $n$are couples.

They are randomly divided into $n$pairs.

Here, $W_i$ is the event.

Step 11: Explanation (Part d)

If the pairing must consist of a men and women, then this experiment reduces to match problem.  

Step 12 : Final Answer (Part b)

The experiment reduces to match problem.