UPSC NDA I 2025 Mathematics Paper

Given,
‌P(A)=‌

1

3

‌P(B)=‌

1

2

‌P(A∩B)=‌

1

4

We need to calculate P(AC∩BC), the probability that neither A nor B occurs.
Using the complement rule:
P(AC∩BC)=1−P(A∪B)
Now, apply the inclusion-exclusion formula to find (P(A∪B) :
P(A∪B)=P(A)+P(B)−P(A∩B)
Substitute the given values:
P(A∪B)=‌

1

3

+‌

1

2

−‌

1

4

To simplify:
P(A∪B)=‌

4

12

+‌

6

12

−‌

3

12

=‌

7

12

Now, calculate P(AC∩BC) :
P(AC∩BC)=1−‌

7

12

=‌

5

12