Question: Membership selection. A town council has 13 members, 5 Democrats and 8 Republicans. (A) If the…

Membership selection. A town council has 13 members, 5 Democrats and 8 Republicans. (A) If the president and vice-president are selected at random, what is the probability that they are both Democrats? (B) If a 3-person committee is selected at random, what is the probability that Republicans make up the majority?

(A) The probability that they are both Democrats is approximately . (Type a decimal. Round to three decimal places.)

Solution

To solve part (A), we want to find the probability that both the president and vice-president selected at random are Democrats. Start with calculating the total number of ways to choose 2 members from 13 members: \[ \binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 78 \] Next, calculate the number of ways to choose 2 Democrats from 5 Democrats: \[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \] Now, find the probability that both are Democrats: \[ P(\text{both are Democrats}) = \frac{\binom{5}{2}}{\binom{13}{2}} = \frac{10}{78} \approx 0.128 \] Thus, the probability that they are both Democrats is approximately 0.128.