CUET UG 2025 — Mathematics Algebra

Statement (A): If E and F are independent events, then $P(E \cap F) = P(E) \cdot P(F)$P(E∩F)=P(E)⋅P(F)

This is the definition of independent events — correct.

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Statement (B): If E and F are mutually exclusive, then $P(E \cup F) = P(E) + P(F) - P(E) \cdot P(F)$P(E∪F)=P(E)+P(F)−P(E)⋅P(F)

Mutually exclusive means $P(E \cap F) = 0$P(E∩F)=0

Using the addition formula:

$$P(E \cup F) = P(E) + P(F) - P(E \cap F)$$P(E∪F)=P(E)+P(F)−P(E∩F)

$$= P(E) + P(F) - 0$$=P(E)+P(F)−0

$$= P(E) + P(F)$$=P(E)+P(F)

The formula given in (B) is the union formula for independent events, not mutually exclusive events — incorrect.

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Statement (C): $P(E|F) = \dfrac{P(E \cap F)}{P(F)},\; P(F) \neq 0$P(E∣F)=P(F)P(E∩F)​,P(F)=0

This is the definition of conditional probability — correct.

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Statement (D): $P(\overline{E}|F) = 2 - P(E|F)$P(E∣F)=2−P(E∣F)

Since E and $\overline{E}$E together cover the entire sample space:

$$P(E|F) + P(\overline{E}|F) = 1$$P(E∣F)+P(E∣F)=1

$$P(\overline{E}|F) = 1 - P(E|F)$$P(E∣F)=1−P(E∣F)

The statement uses $2$2 instead of $1$1 — incorrect.

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Statements (A) and (C) are correct.