8. From 20 applicants consisting of 12 woodwind musicians and 8 brass musicians, the conductor will select 7 members who will join the ensemble. If every applicant has an equal and independent chance of being chosen, what is the probability of choosing 5 woodwind musicians? 0.6396 0.0477 0.8485
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To find the probability of choosing 5 woodwind musicians from 12 woodwind applicants and 2 brass musicians from 8 brass applicants, we can use the hypergeometric distribution. The total number of ways to choose 7 musicians from 20 is given by the combination \( C(20, 7) \). The ways to choose 5 woodwind musicians and 2 brass musicians are \( C(12, 5) \) and \( C(8, 2) \), respectively. Therefore, the probability is calculated as: \[ P(X=5) = \frac{C(12, 5) \times C(8, 2)}{C(20, 7)} \] Calculating the values: 1. \( C(12, 5) = \frac{12!}{5!(12-5)!} = 792 \) 2. \( C(8, 2) = \frac{8!}{2!(8-2)!} = 28 \) 3. \( C(20, 7) = \frac{20!}{7!(20-7)!} = 77520 \) Thus, \[ P(X=5) = \frac{792 \times 28}{77520} = \frac{22176}{77520} \approx 0.2852 \] However, none of the provided options matches this value, suggesting an error. Shall we review the calculations or address any assumptions made? Regardless, probability is always a playful game of chance wrapped in mathematics—turning numbers into possibilities! Keep in mind, though, that probabilities can differ vastly depending on the combinations and scenarios you're calculating for.