Question

a city council consists of 10 members. four are republicans, three are democrats, and three are independents. if a committee of three is to be selected, find the probability of selecting
(a) all republicans. round your answer to five decimal places.
the probability of selecting all republicans is

Explanation:

Step1: Calculate the number of ways to choose 3 Republicans from 4 Republicans

Use the combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\), where \(n = 4\) (number of Republicans) and \(k=3\) (number of members to be chosen).

$$C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=\frac{4\times3!}{3!×1}=4$$

Step2: Calculate the number of ways to choose 3 members from 10 members

Here \(n = 10\) (total number of council members) and \(k = 3\) (number of members to be chosen).

$$C(10,3)=\frac{10!}{3!(10 - 3)!}=\frac{10\times9\times8\times7!}{3\times2\times1\times7!}=120$$

Step3: Calculate the probability

The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).

$$P=\frac{C(4,3)}{C(10,3)}=\frac{4}{120}\approx0.03333$$

Answer:

\(0.03333\)