Question

consider a political discussion group consisting of 10 democrats, 9 republicans, and 3 independents. suppose that two group members are randomly selected, in succession, to attend a political convention. find the probability of selecting a republican and then a democrat
(type an integer or a simplified fraction)

Explanation:

Step1: Calculate total number of members

Total members = \(10 + 9+3=22\)

Step2: Calculate probability of selecting a Republican first

Probability of selecting a Republican first \(P(R)=\frac{9}{22}\)

Step3: Calculate probability of selecting a Democrat second

After selecting a Republican, number of members left = \(21\), number of Democrats = \(10\). Probability of selecting a Democrat second \(P(D|R)=\frac{10}{21}\)

Step4: Calculate the combined probability

Using the multiplication rule for dependent events \(P(R\cap D)=P(R)\times P(D|R)\)
\(P(R\cap D)=\frac{9}{22}\times\frac{10}{21}=\frac{9\times10}{22\times21}=\frac{90}{462}=\frac{15}{77}\)

Answer:

\(\frac{15}{77}\)