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. (i) Find the probability that student opted for NCC or NSS. (ii) Probability that the student has opted neither NCC nor NSS.

Explanation

To solve the problem step by step, we will use the principles of probability and set theory.

### Given:
- Total number of students (N) = 60
- Number of students who opted for NCC (A) = 30
- Number of students who opted for NSS (B) = 32
- Number of students who opted for both NCC and NSS (A ∩ B) = 24

### Step 1: Find the probability that a student opted for NCC or NSS (A ∪ B)

We will use the formula for the probability of the union of two events:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

#### Step 1.1: Calculate P(A)

\[ P(A) = \frac{\text{Number of students who opted for NCC}}{\text{Total number of students}} = \frac{30}{60} = \frac{1}{2} \]

#### Step 1.2: Calculate P(B)

\[ P(B) = \frac{\text{Number of students who opted for NSS}}{\text{Total number of students}} = \frac{32}{60} = \frac{8}{15} \]

#### Step 1.3: Calculate P(A ∩ B)

\[ P(A \cap B) = \frac{\text{Number of students who opted for both NCC and NSS}}{\text{Total number of students}} = \frac{24}{60} = \frac{2}{5} \]

#### Step 1.4: Substitute values into the union formula

Now substituting the values into the union formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{1}{2} + \frac{8}{15} - \frac{2}{5} \]

To perform this calculation, we need a common denominator. The least common multiple of 2, 15, and 5 is 30.

Convert each fraction:

\[ \frac{1}{2} = \frac{15}{30}, \quad \frac{8}{15} = \frac{16}{30}, \quad \frac{2}{5} = \frac{12}{30} \]

Now substitute:

\[ P(A \cup B) = \frac{15}{30} + \frac{16}{30} - \frac{12}{30} = \frac{19}{30} \]

### Step 2: Find the probability that a student has opted neither NCC nor NSS

To find this probability, we can use the complement rule:

\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \]

Substituting the value we found:

\[ P(\text{neither A nor B}) = 1 - \frac{19}{30} = \frac{30}{30} - \frac{19}{30} = \frac{11}{30} \]

### Final Answers:
(i) The probability that a student opted for NCC or NSS is \( \frac{19}{30} \).

(ii) The probability that the student has opted neither NCC nor NSS is \( \frac{11}{30} \).

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