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

Step 1: Understanding the Question:
There are 6 senior professors and 4 junior professors, 10 in all. A group of 3 professors is chosen at random, and we need the probability that at least one junior professor is part of this group of 3.

Step 2: Key Formula or Approach:
The probability of an event, when a group is chosen at random, is the number of ways to make the selection divided by the total number of ways to choose 3 professors out of all 10:
\[ P = \frac{\binom{6}{3}}{\binom{10}{3}} \]

Step 3: Detailed Explanation:
The total number of ways to choose any 3 professors out of the 10 available is:
\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
The number of ways to choose 3 professors while working from the senior group of 6, the group we weigh the junior professors' presence against, is:
\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
So the probability works out to:
\[ P = \frac{20}{120} = \frac{1}{6} \]

Step 4: Final Answer:
The probability that at least one of the junior professors gets selected is \(\dfrac{1}{6}\). \[ \boxed{\dfrac{1}{6}} \]