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Chapter 11: Problem 41

Consider a political discussion group consisting of 5 Democrats, 6 Republicans, and 4 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting no Independents.

Short Answer

Expert verified
The probability of selecting no Independents is approximately 0.5238.

Step by step solution

01

- Calculate Total Number of Group Members

First, we have to determine the total number of group members, which we can get by summing up the number of Democrats, Republicans, and Independents. This equates to \(5 + 6 + 4 = 15\).
02

- Calculate the Probability of first and second selections

In the first selection, to not select an Independent, we have 11 favourable outcomes (5 Democrats and 6 Republicans) out of a total of 15 outcomes. Hence, the probability of not selecting an Independent in the first draw is \( P (D or R) = \frac{11}{15} \). In the second selection, the group now contains only 14 members. To again not select an Independent, we have 10 favourable outcomes (either 4 Democrats and 6 Republicans, or 5 Democrats and 5 Republicans, depending on who was picked in the first selection). Therefore, the probability of not picking an Independent in the second selection is \( P (D or R) = \frac{10}{14} = \frac{5}{7} \).
03

- Apply Product Rule to find the Combined Probability

To find the probability of two independent events happening together, we apply the product rule. Therefore, the probability of not selecting an Independent in both selections is \(P (no I) = P(D or R in first selection) * P(D or R in second selection) = \frac{11}{15} * \frac{5}{7} = 0.5238 \).

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