CUET UG 2025 — Mathematics Algebra

For independent events E and F, one event happening doesn't affect the probability of the other event happening.

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The fundamental property of independent events is:

$$P(E \cap F) = P(E) \times P(F)$$P(E∩F)=P(E)×P(F)

The probability of both events occurring together equals the product of their individual probabilities.

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Using the conditional probability formula:

$$P(E|F) = \frac{P(E \cap F)}{P(F)}$$P(E∣F)=P(F)P(E∩F)​

Substituting the independence property:

$$P(E|F) = \frac{P(E) \times P(F)}{P(F)}$$P(E∣F)=P(F)P(E)×P(F)​

$$P(E|F) = P(E)$$P(E∣F)=P(E)

This shows that even after F occurs, the probability of E remains unchanged, which is the definition of independence.

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Option 1: $P(E \cap F) = 0$P(E∩F)=0 would mean E and F are mutually exclusive, not independent.

Option 2: $P(E \cup F) = 1$P(E∪F)=1 would mean E or F always occurs, which independence doesn't guarantee.

Option 4: $P(E|F) = P(F)$P(E∣F)=P(F) is incorrect as $P(E|F)$P(E∣F) relates to event E, not event F.

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The correct statement is Option 3: $P(E|F) = P(E)$P(E∣F)=P(E)