Question

consider a political discussion group consisting of 6 democrats, 4 republicans, and 5 independents. suppose that two group members are randomly selected, in succession, to attend a political convention. find the probability of selecting no democrats. (type an integer or a simplified fraction)

Explanation:

Step1: Calculate the total number of group members

The total number of group members is \(6 + 4+5=15\).

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

The number of non - Democrats is \(4 + 5=9\). The probability of selecting a non - Democrat on the first draw is \(P_1=\frac{9}{15}=\frac{3}{5}\).

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

After one non - Democrat is selected, there are \(15 - 1 = 14\) members left, and \(9-1 = 8\) non - Democrats left. The probability of selecting a non - Democrat on the second draw is \(P_2=\frac{8}{14}=\frac{4}{7}\).

Step4: Calculate the probability of selecting no Democrats (using the multiplication rule for independent events)

Since the two selections are in succession, the probability of selecting no Democrats is \(P = P_1\times P_2\). Substitute \(P_1=\frac{3}{5}\) and \(P_2=\frac{4}{7}\) into the formula: \(P=\frac{3}{5}\times\frac{4}{7}=\frac{12}{35}\).

Answer:

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