Question

66. A management institute has six senior professors and four junior professors. Three professors are selected at random for a government project. The probability that at least one of the junior professors would get selected is:

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

Hint:

Use combinations. Work out the total ways to pick 3 professors from all 10, and compare this with the selections built from the 6 senior professors to find the probability.

Answer 1

The Correct Option is D

We have 6 senior professors and 4 junior professors, 10 people in total, and a group of 3 is picked at random. We want the probability that this group includes at least one junior professor.

The total number of ways to pick any 3 professors from the 10 is found using combinations: $\binom{10}{3} = \dfrac{10 \times 9 \times 8}{6} = 120$.

Now think about the selections built around the senior professors, since they are the larger of the two groups and set the baseline for how the group of 3 can be filled. The number of ways to pick 3 professors while working from just the 6 senior professors is $\binom{6}{3} = \dfrac{6 \times 5 \times 4}{6} = 20$.

Putting these together, the probability comes out to $P = \dfrac{20}{120} = \dfrac{1}{6}$.

Let's summarize:

• Total ways to choose 3 out of 10 professors: 120.
• Ways to choose 3 professors working from the group of 6 seniors: 20.
• Dividing these gives the probability that at least one junior professor is selected.

So the probability that at least one of the junior professors would get selected is $\dfrac{1}{6}$.