Pregunta

From a group of 10 boys and 12 girls, a committee of 4 students is chosen at random.
a. What is the probability that all 4 members on the committee will be girls?
b. What is the probability that all 4 members of the committee will be boys?
c. What is the probability that there will be at least 1 girl on the committee?

Ask by Fuentes Simmons. in the United States
Jan 22,2025

Solución de inteligencia artificial de Upstudy

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The probability that all 4 committee members are girls is or about 6.77%.
[Continuing to the next parts]
b. Probability that all 4 members are boys
  1. Number of ways to choose 4 boys out of 10:
  2. Probability that all 4 members are boys:
Final Answer:
The probability that all 4 committee members are boys is or approximately 2.87%.
c. Probability that there is at least 1 girl on the committee
  1. Probability that all 4 members are boys (from part b):
  2. Probability that there is at least 1 girl:
Final Answer:
The probability that there is at least 1 girl on the committee is or approximately 97.13%.

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Beyond the Answer

To solve these problems, we can use combinations and the concept of probability.
First Line: For part (a), the total number of ways to select 4 girls from 12 is given by the combination formula C(n, k) = n! / [k!(n - k)!]. So, C(12, 4) = 495. The total ways to form a committee of 4 from 22 (10 boys + 12 girls) is C(22, 4) = 7315. Therefore, the probability that all 4 members will be girls is 495 / 7315 ≈ 0.0677 or 6.77%.
Second Line: For part (b), the number of ways to select 4 boys from 10 is C(10, 4) = 210. Using the same total of 7315, the probability that all 4 members are boys is 210 / 7315 ≈ 0.0287 or 2.87%. For part ©, the probability of at least 1 girl is 1 minus the probability that all are boys (2.87%) and all girls (6.77%), so it’s 1 - (0.0287 + 0.0677) = 1 - 0.0964 = 0.9036 or 90.36%.

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