Question 9 ( 5 points)
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Solve the problem that involves probabilities with events that are not mutually exclusive.
Consider a political discussion group consisting of 6 Democrats, 3 Republicans, and 5 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting an Independent and then a Democrat.
A) $\frac{15}{98}$
B) $\frac{3}{91}$
C) $\frac{5}{182}$
D) $\frac{15}{91}$
Answer
Simplify the fraction if possible: \(\frac{15}{91}\)
Step 1 ：Calculate the total number of ways to select two group members from the political discussion group using the combination formula: \(C(14, 2) = 91\)
Step 2 ：Calculate the number of ways to select an Independent and then a Democrat: \(C(5, 1) \times C(6, 1) = 30\)
Step 3 ：Calculate the probability by dividing the number of ways to select an Independent and then a Democrat by the total number of ways to select two group members: \(\frac{30}{91}\)
Step 4 ：Simplify the fraction if possible: \(\frac{15}{91}\)
