Question

A management institute has 6 senior professors and 4 junior professors, 3 professors are selected at random for a government project. The probability that at least one of the junior professors would get selected is :

• A. \(\frac{5}{6}\)
• B. \(\frac{2}{3}\)
• C. \(\frac{1}{5}\)
• D. \(\frac{1}{6}\)

Answer 1

The Correct Option is A

To solve this problem, we need to find the probability that at least one junior professor is selected when 3 professors are chosen from a group of 6 senior professors and 4 junior professors.

Let's break down the solution step-by-step:

1. Calculate the total number of ways to select 3 professors from the 10 professors (6 seniors + 4 juniors):
2. \(\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120\)
3. Calculate the number of ways to select 3 professors when none of them are junior (i.e., all are senior professors):
4. \(\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\)
5. Use the complementary probability approach. The probability that at least one junior professor is selected is 1 minus the probability that no junior professor is selected:
6. The probability that no junior professors are selected (only senior professors are selected) is:
7. P(\text{no junior}) = \frac{\binom{6}{3}}{\binom{10}{3}} = \frac{20}{120} = \frac{1}{6}
8. Therefore, the probability that at least one junior professor is selected is:
9. P(\text{at least one junior}) = 1 - P(\text{no junior}) = 1 - \frac{1}{6} = \frac{5}{6}

Hence, the probability that at least one junior professor would get selected is \(\frac{5}{6}\).

Correct option: \(\frac{5}{6}\).