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3-person committee is to be chosen from 13 British people and 11 Americans. What is the probability that the committee will consist of all Americans? The events of repeatedly selecting a person from 13 British people and 11 Americans, if the same person cannot be selected twice, are? Probability: (Simplify your answer and round to four decimal places.)

3-person committee is to be chosen from 13 British people and 11 Americans. What is the probability that the committee will consist of all Americans? The events of repeatedly selecting a person from 13 British people and 11 Americans, if the same person cannot be selected twice, are? Probability: (Simplify your answer and round to four decimal places.)

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 7ECP: You draw one card at random from a standard deck of 52 playing cards. What is the probability that...
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### Probability Problem: Selecting an All-American Committee **Problem Statement:** A 3-person committee is to be chosen from 13 British people and 11 Americans. What is the probability that the committee will consist of all Americans? **Text Explanation:** The events of repeatedly selecting a person from 13 British people and 11 Americans, if the same person cannot be selected twice, are represented below. **Probability Calculation:** Probability: (Simplify your answer and round to four decimal places.) --- ### Detailed Explanation for Understanding: To find the probability that the committee will consist of all Americans, we need to carefully count the number of ways to select the committee from the Americans and divide it by the total number of ways to select the committee from the entire group. 1. **Total people:** - 13 British - 11 Americans - Total = 13 + 11 = 24 people 2. **Selecting all Americans:** - We need to select 3 Americans out of 11. - The number of ways to choose 3 people out of 11 is given by the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) Thus, \( \binom{11}{3} = \frac{11!}{3!(11-3)!} = \frac{11!}{3! \cdot 8!} \) 3. **Total number of ways to select 3 people out of 24:** - Similarly, the total number of ways to choose 3 people out of 24 is given by \( \binom{24}{3} = \frac{24!}{3!(24-3)!} = \frac{24!}{3! \cdot 21!} \) 4. **Probability Calculation:** - The probability is the ratio of the two combinations calculated above. - Probability \( = \frac{\binom{11}{3}}{\binom{24}{3}} \) 5. **Simplification:** - Perform the calculations to simplify and find the probability, and then round it to four decimal places. By following these steps, the exact value can be obtained and will help in understanding the probability of selecting an all-American committee.
Transcribed Image Text:### Probability Problem: Selecting an All-American Committee **Problem Statement:** A 3-person committee is to be chosen from 13 British people and 11 Americans. What is the probability that the committee will consist of all Americans? **Text Explanation:** The events of repeatedly selecting a person from 13 British people and 11 Americans, if the same person cannot be selected twice, are represented below. **Probability Calculation:** Probability: (Simplify your answer and round to four decimal places.) --- ### Detailed Explanation for Understanding: To find the probability that the committee will consist of all Americans, we need to carefully count the number of ways to select the committee from the Americans and divide it by the total number of ways to select the committee from the entire group. 1. **Total people:** - 13 British - 11 Americans - Total = 13 + 11 = 24 people 2. **Selecting all Americans:** - We need to select 3 Americans out of 11. - The number of ways to choose 3 people out of 11 is given by the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) Thus, \( \binom{11}{3} = \frac{11!}{3!(11-3)!} = \frac{11!}{3! \cdot 8!} \) 3. **Total number of ways to select 3 people out of 24:** - Similarly, the total number of ways to choose 3 people out of 24 is given by \( \binom{24}{3} = \frac{24!}{3!(24-3)!} = \frac{24!}{3! \cdot 21!} \) 4. **Probability Calculation:** - The probability is the ratio of the two combinations calculated above. - Probability \( = \frac{\binom{11}{3}}{\binom{24}{3}} \) 5. **Simplification:** - Perform the calculations to simplify and find the probability, and then round it to four decimal places. By following these steps, the exact value can be obtained and will help in understanding the probability of selecting an all-American committee.
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