Q.In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that

Concept: Set Operations and Probability

We have $n(\text{NCC}) = 30$, $n(\text{NSS}) = 32$, $n(\text{NCC} \cap \text{NSS}) = 24$, and total students $= 60$.

Step 1. Students who opted for NCC or NSS:

So $P(\text{NCC or NSS}) = \frac{38}{60} = \frac{19}{30}$.

Step 2. Students who opted for neither:

So $P(\text{Neither}) = \frac{22}{60} = \frac{11}{30}$.

Step 3. Students who opted for NSS but not NCC: …

Understanding Set Operations and Probability

When students opt for activities like NCC or NSS, we're dealing with overlapping sets. Some students choose one, some choose the other, and some choose both. The inclusion-exclusion principle tells us how to count without double-counting those who opted for both.

The key insight: if 30 students opted for NCC and 32 for NSS, we can't simply add these to get the total who opted for at least one activity, because the 24 students who opted for both would be counted twice. We must subtract the overlap once.

For probability, we then divide the number of favorable outcomes by the total number of students.

Setting Up the Problem

Let's denote:

• $n(N) = 30$ (students who opted for NCC)
• $n(S) = 32$ (students who opted for NSS)
• $n(N \cap S) = 24$ (students who opted for both)
• Total students $= 60$

This is the inclusion-exclusion principle for two sets.

Solution

1. Students who opted for NCC or NSS

Using inclusion-exclusion:

So 38 students opted for at least one of the two activities.

The probability is:

2. Students who opted for neither NCC nor NSS

If 38 students opted for at least one activity, then the remaining students opted for neither:

The probability is:

3. Students who opted for NSS but not NCC …