In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If

Q: Wait, why do we subtract the intersection? Can't we just add the two groups together?

That is a classic trap! If you simply add $n(A)$ and $n(B)$, you are counting the students who chose *both* activities twice—once when you count the NCC group, and again when you count the NSS group. To get an accurate count of unique students, we must subtract that overlap once to 'fix' the double counting. It is the fundamental logic of the Inclusion-Exclusion Principle.

Q: Is the 'Universal Set' always the total number of students in the class?

In this specific problem, yes. The Universal Set, denoted as $S$, represents the entire sample space—every single student we are considering. When we talk about probability, the denominator is always the size of this Universal Set. Always define your 'Universe' first before you start carving it up into subsets!

Q: What if the question asked for 'only NCC' instead of 'neither'? How would the formula change?

Great question! If you need 'only NCC,' you are looking for the region inside the NCC circle that does *not* overlap with the NSS circle. Mathematically, that is $n(A) - n(A \cap B)$. You take the total NCC group and remove the students who also chose NSS. Always visualize the Venn diagram; it makes these logical subtractions intuitive.

Question

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :

Options:

(A) $2/3$

(B) $1/6$

(C) $1/3$

(D) $5/6$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a classroom of $60$ students. This is your reality. In set theory, we call this the **Universal Set**, denoted as $S$.

Everything we do, every calculation we perform, must exist within the boundaries of this $60$. We define our sample space as $n(S) = 60$. This is the foundation upon which we build our probability.

### The Overlapping Reality

Now, let us introduce the two activities: the National Cadet Corps (NCC) and the National Service Scheme (NSS). We represent these as two sets, $A$ and $B$. We are given $n(A) = 40$ and $n(B) = 30$.

Most students mistakenly assume the total is $40 + 30 = 70$. However, since $70 > 60$, we must account for the intersection. The problem states that $20$ students opted for *both* NCC and NSS, meaning $n(A \cap B) = 20$.

### The Inclusion-Exclusion Principle

To find the number of students who opted for *at least one* activity, we use the **Inclusion-Exclusion Principle**. This formula corrects for the double-counting of students who belong to both groups:

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Substituting our known values into the equation:

$$n(A \cup B) = 40 + 30 - 20$$

$$n(A \cup B) = 50$$

Thus, $50$ students are active in at least one program. They are the ones who have "stepped into the circles" of our Venn diagram.

### Finding the 'Neither' Region

Now, we look for the students who opted for *neither*. These are the students who are sitting in the classroom but are not part of the NCC or NSS circles.

To find them, we subtract the active students from the total population:

$$n(	ext{Neither}) = n(S) - n(A \cup B)$$

$$n(	ext{Neither}) = 60 - 50 = 10$$

There are exactly $10$ students who have chosen neither path. These individuals represent the complement of our union.

### The Final Probability

Probability is the ratio of favorable outcomes to total possible outcomes. We want to find the probability of selecting one of the $10$ students who chose neither.

$$P(	ext{Neither}) = \frac{n(	ext{Neither})}{n(S)}$$

$$P(	ext{Neither}) = \frac{10}{60}$$

By simplifying this fraction, we arrive at our final result:

**$$P(	ext{Neither}) = \frac{1}{6}$$**

### The Takeaway

We didn't just plug numbers into a formula; we visualized the classroom, identified the double-counting trap, and corrected it. Whether you are dealing with sets, vectors, or complex numbers, always start by visualizing the "Universe" and understanding the relationships between your variables. You have mastered this concept today.

Animated Solution for Mathematics - Probability: In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :

Select Answer:

Visualized Solution

Universal Set $n (S)$

Set $A$ : NCC Students

Set $B$ : NSS Students

Intersection $n (A \cap B)$

The Union Formula

Substitute Values

Calculate $n (A \cup B)$

Identify 'Neither' Students

Calculate 'Neither' Count

Probability Formula

Final Calculation

The Sigma Insight: Classical Definition of Probability

Analyzing the Setup

The Overlapping Reality

The Inclusion-Exclusion Principle

Finding the 'Neither' Region

The Final Probability

The Takeaway

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