Q. 2.49 - A First Course in Probability

Question

A group of $6$men and $6$women is randomly divided into $2$groups of size $6$each. What is the probability that both groups will have the same number of men?

Step-by-Step Solution

Answer

the probability that both groups will have the same number of men is $0.4329$

Step 1 Given Information.

A group of $6$men and $6$women is randomly divided into $2$groups of size $6$each.

Step 2 Explanation.

 Division of $6$men and $6$women into two groups.

The outcome space $S$contains all equally likely outcomes - every $6$person subgroup from this group of $12$people.

 $\left(\begin{array}{c}12\\ 6\end{array}\right)$

And there are $\left(\begin{array}{l}6\\ 3\end{array}\right)$ways to choose the three men for one group, and $\left(\begin{array}{l}6\\ 3\end{array}\right)$possible choices for women. This makes:

${\left(\begin{array}{l}6\\ 3\end{array}\right)}^2$

Since the sample outcomes are equally likely:

$P\left(\text{ groups have }3\text{ men }\right)=\frac{\text{ \# groups with three men }}{\text{ \# groups in the outcome space }}=\frac{{\left(\begin{array}{c}6\\ 3\end{array}\right)}^2}{\left(\begin{array}{c}12\\ 6\end{array}\right)}=0.4329\dots$