CUET UG 2025 — Mathematics Algebra

Given:

$$P(A) = 0.45$$P(A)=0.45

$$P(B) = 0.20$$P(B)=0.20

$$P(A \cap B) = 0.06$$P(A∩B)=0.06

The probability that neither A nor B occurs is the complement of the probability that at least one of them occurs.

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Using the addition rule of probability:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$P(A∪B)=P(A)+P(B)−P(A∩B)

$$P(A \cup B) = 0.45 + 0.20 - 0.06$$P(A∪B)=0.45+0.20−0.06

$$P(A \cup B) = 0.65 - 0.06$$P(A∪B)=0.65−0.06

$$P(A \cup B) = 0.59$$P(A∪B)=0.59

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The probability that neither A nor B occurs:

$$P(\text{neither A nor B}) = 1 - P(A \cup B)$$P(neither A nor B)=1−P(A∪B)

$$P(\text{neither A nor B}) = 1 - 0.59$$P(neither A nor B)=1−0.59

$$P(\text{neither A nor B}) = 0.41$$P(neither A nor B)=0.41

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Therefore, the probability that neither A nor B occurs is $0.41$0.41.