Questions: A city council consists of six Democrats and eight Republicans. If a committee of five people is selected, find the probability of selecting two Democrats and three Republicans. (Type a fraction. Simplify your answer.)

Solution Steps

To find the probability of selecting exactly 2 Democrats from a total of 6 Democrats, we use the hypergeometric distribution formula:

\[ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \]

Substituting the values:

• \( N = 14 \) (total members)
• \( K = 6 \) (total Democrats)
• \( n = 5 \) (committee size)
• \( k = 2 \) (Democrats to select)

We calculate:

\[ P(X = 2) = \frac{\binom{6}{2} \binom{8}{3}}{\binom{14}{5}} = 0.4196 \]

Next, we calculate the probability of selecting exactly 3 Republicans from a total of 8 Republicans using the same hypergeometric distribution formula:

Substituting the values:

• \( N = 14 \) (total members)
• \( K = 8 \) (total Republicans)
• \( n = 5 \) (committee size)
• \( k = 3 \) (Republicans to select)

We calculate:

\[ P(X = 3) = \frac{\binom{8}{3} \binom{6}{2}}{\binom{14}{5}} = 0.4196 \]

The probability of selecting 2 Democrats and 3 Republicans is the product of the two probabilities calculated above:

\[ P(X = 2 \text{ Democrats and } 3 \text{ Republicans}) = P(X = 2) \times P(X = 3) = 0.4196 \times 0.4196 = 0.1761 \]

Final Answer