Question

consider a political discussion group consisting of 6 democrats, 6 republicans, and 3 independents. suppose that two group members are randomly selected, in succession to attend a political convention. find the probability of selecting no republicans. (type an integer or a simplified fraction)

Explanation:

Step1: Calculate the total number of members

The total number of members is \(6 + 6+3=15\).

Step2: Calculate the probability of the first - non - Republican selection

The number of non - Republicans is \(6 + 3=9\). The probability of selecting a non - Republican first is \(P_1=\frac{9}{15}=\frac{3}{5}\).

Step3: Calculate the probability of the second - non - Republican selection (after one non - Republican is already selected)

After one non - Republican is selected, the number of non - Republicans left is \(9 - 1 = 8\), and the total number of members left is \(15-1 = 14\). The probability of selecting a non - Republican second is \(P_2=\frac{8}{14}=\frac{4}{7}\).

Step4: Calculate the combined probability

Since the two selections are independent events (in the sense of sequential without replacement), the probability of selecting no Republicans (i.e., two non - Republicans) is \(P = P_1\times P_2\).

$$P=\frac{3}{5}\times\frac{4}{7}=\frac{12}{35}$$

Answer:

\(\frac{12}{35}\)