Chapter 11.6, Problem 21PPS

To determine

Find the probability that 7 of the members will be women.

Answer to Problem 21PPS

The probability that 7 of the members are women is $\frac{15}{128}\approx 0.117$

Explanation of Solution

Given:

If an equal number of men and women applied to be on a community planning committee and the committee needs a total of 10 people, find the probability that 7 of the members will be women. Assume that committee members will be chosen randomly.

Concept Used:

Find the probability by expanding ${(m+w)}^{10}$

Calculation:

  $\begin{array}{l}$ {(m+w)}^{10}=m^{10}+10m^{9}w+45m^8{w}^2+120m^7{w}^3+$210m^6{w}^4+252m^5{w}^5+210m^4{w}^6+\\ 120m^3{w}^7+45m^2{w}^8+10m{w}^9+w^{10}\end{array}$

By adding the coefficients of the polynomial, we determine that there are 1024 combinations of men and women.

The term $120m^3{w}^7$ represents the number of combinations with 7 women and 3 men.

Therefore, the probability that 7 of the members are women is $\frac{120}{1024}=\frac{15}{128}\approx 0.117$

Thus, the probability that 7 of the members are women is $\frac{15}{128}\approx 0.117$