If P(A)=1/3,P(B)=1/2 and P(A∩B)=1/4, then what is the value of P(A^{C}∩B^{C})?

Correct Answer is : 5/125/125/12

Correct Answer is : 5/125/125/12

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UPSC NDA I 2025 Mathematics Paper

Question : 116 of 120

Marks: +1, -0

P(A)=1/3,P(B)=1/2

P(A∩B)=1/4

P(A^{C}∩B^{C})

Solution:

Concept:

Use De Morgan's law:

P(A^c \cap B^c) = 1 - P(A \cup B)

The probability that neither event occurs equals one minus the probability that at least one occurs.

Explanation:

First, find

P(A \cup B)

using the addition rule:

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

Substitute the given values:

P(A \cup B) = \frac{1}{3} + \frac{1}{2} - \frac{1}{4}

Convert to a common denominator of 12:

\frac{4}{12} + \frac{6}{12} - \frac{3}{12} = \frac{7}{12}

Now apply the complement rule:

P(A^c \cap B^c) = 1 - \frac{7}{12} = \frac{5}{12}

Answer:

Option B:

\frac{5}{12}

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