The problem asks us to find the probability that a group of 13 people selected from 4 freshmen, 3 sophomores, 6 juniors, and 5 seniors has at least 2 sophomores.

Total number of students:

• Freshmen: 4
• Sophomores: 3
• Juniors: 6
• Seniors: 5

Total students = $4 + 3 + 6 + 5 = 18$

Step 1: Total possible selections

We are choosing 13 people from a pool of 18, so the total number of ways to choose 13 students from 18 is: $\binom{18}{13}$

Step 2: Find complementary probability (no sophomores or only 1 sophomore)

To find the probability of having at least 2 sophomores, we can first find the complementary probability (probability of having 0 or 1 sophomore) and subtract it from 1.

Case 1: No sophomores

If no sophomores are selected, we are choosing all 13 students from the 15 non-sophomores (4 freshmen + 6 juniors + 5 seniors), so the number of ways to do this is: $\binom{15}{13}$

Case 2: One sophomore

If exactly one sophomore is selected, we first choose 1 sophomore from the 3 available sophomores and then choose the remaining 12 students from the 15 non-sophomores. The number of ways to do this is: $\binom{3}{1} \times \binom{15}{12}$

Step 3: Total number of unfavorable selections (0 or 1 sophomore)

The total number of unfavorable selections is the sum of the two cases: $\text{Total unfavorable} = \binom{15}{13} + \binom{3}{1} \times \binom{15}{12}$

Step 4: Probability of at least 2 sophomores

Now, the probability of having at least 2 sophomores is: $P(\text{at least 2 sophomores}) = 1 - \frac{\text{Total unfavorable}}{\binom{18}{13}}$

Let me calculate this.The probability that the group of 13 people has at least 2 sophomores is approximately 0.8284 (rounded to four decimal places).

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the probability change if the committee required 14 people instead of 13?
2. What is the probability of selecting exactly 3 sophomores for the same group size of 13?
3. How would you calculate the probability if juniors were the focus instead of sophomores?
4. What is the probability of selecting at least 1 sophomore if the group size were reduced to 10?
5. How would the total number of possible groups change if only juniors and seniors were considered?

Tip: When calculating complementary probabilities, it's often easier to solve for the opposite event and subtract it from 1.