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Difficulty: HardBasic Probability and Counting Methods

A music ensemble director must randomly select 3 students from a group consisting of 5 vocalists and 4 instrumentalists to perform at a ceremony. What is the probability that the chosen group contains at least 1 vocalist and at least 1 instrumentalist?

  1. A
    16\frac{1}{6}
  2. B
    1021\frac{10}{21}
  3. C
    521\frac{5}{21}
  4. D
    2081\frac{20}{81}
  5. 56\frac{5}{6}Answer

Answer

56\frac{5}{6}
To find the probability that the selected trio includes at least 1 vocalist and at least 1 instrumentalist, calculate the total possible combinations of picking 3 students from 9: (93)=84\binom{9}{3} = 84. Next, identify the unfavorable combinations where all 3 students are of the same type: (53)=10\binom{5}{3} = 10 ways for all vocalists and (43)=4\binom{4}{3} = 4 ways for all instrumentalists, totaling 10+4=1410 + 4 = 14 unfavorable outcomes. Subtracting these from 84 leaves 70 favorable combinations. Thus, the probability is 7084=56\frac{70}{84} = \frac{5}{6}.

Step-by-Step Solution

1
Calculate the total number of ways to choose any 3 students from the total group of 9 students.
(93)=9×8×73×2×1=84\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 total outcomes.
Establishes the size of the sample space for selecting 3 students without replacement out of 9 (5 vocalists + 4 instrumentalists).
2
Determine the number of unfavorable outcomes (groups consisting entirely of one type of student).
Ways to choose 3 vocalists: (53)=10\binom{5}{3} = 10.
Ways to choose 3 instrumentalists: (43)=4\binom{4}{3} = 4.
Total unfavorable outcomes = 10+4=1410 + 4 = 14.
The complement of having 'at least 1 vocalist and at least 1 instrumentalist' is selecting a group of 3 composed entirely of vocalists or entirely of instrumentalists.
3
Subtract the unfavorable outcomes from the total outcomes to get the favorable outcomes.
84−14=7084 - 14 = 70 favorable combinations.
Finding the complementary set count is direct and prevents missing valid composition cases.
4
Calculate the probability by dividing favorable outcomes by total outcomes.
Probability = 7084=56\frac{70}{84} = \frac{5}{6}.
Dividing favorable combinations by the total size of the sample space gives the required probability.

Key Concept

Complementary Probability and Combinations
Estimated Time:1m 30s
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Topics

Basic Probability and Counting Methods

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