UGC NET Questions (Paper – 1)

B is the definition of independence: P(A ∩ B) = P(A)P(B). C is correct because mutually exclusive events with positive probabilities have P(A ∩ B)=0 but P(A)P(B)>0, so they cannot be independent. D is true because independence carries over to complements. A is false (addition relates to mutually exclusive unions, not independence). E is false because for independent events P(A|B) = P(A), not 0. Therefore B, C and D only are correct.

Option A:
Option A is incomplete because it includes only B and leaves out C and D, which are also correct statements.

Option B:
Option B is correct as it includes exactly the true statements and excludes A and E, which confuse addition/conditional probability with independence.

Option C:
Option C is wrong because it includes E, which misstates conditional probability under independence.

Option D:
Option D is incorrect because it includes A, which is not the criterion for independence.

Conditional probability is written using a vertical bar “|”. P(A|B) means “the probability of A given that B has already occurred,” i.e., the sample space is restricted to outcomes where B happens.

Option A:
P(A ∩ B) is the probability that both A and B occur together (intersection). It does not represent “given B occurred,” so it is not conditional probability.

Option B:
P(A ∪ B) is the probability that A or B (or both) occur (union). It does not express “given B,” so it is not the required notation.

Option C:
P(A|B) is correct because it explicitly means “probability of A given B.” This is the standard notation for conditional probability in probability theory and statistics.

Option D:
P(B|A) means “probability of B given A,” which reverses the condition. The question asks for “A given B,” so this is not correct.

A defines conditional probability, and B states the law of total probability for a partition. C is true because independence means conditioning on B does not change the probability of A. D is true because Bayes’ theorem updates a prior probability using likelihood/evidence. E is false: P(A∣B) + P(A∣B′) is not generally 1 (they are conditional probabilities on different conditions; their sum can be greater than 1 or less than 1). Therefore A, B, C and D only are correct.

Option A:
Option A is incorrect because it omits D, and Bayes’ theorem is a correct statement commonly paired with conditional probability concepts.

Option B:
Option B is incorrect because it omits A, which is the defining formula for conditional probability.

Option C:
Option C is correct because it includes all and only the true statements (A, B, C and D) and excludes E, which is not a valid identity in general.

Option D:
Option D is incorrect because it omits B, the law of total probability, which is true.

There are 4 aces in a standard 52-card deck. The probability that the first card is an ace is 4/52, since any of the 4 aces could appear among the 52 cards. After one ace has been drawn, 3 aces remain in a reduced deck of 51 cards, so the probability that the second card is also an ace is 3/51. Because the draws are without replacement and the events are dependent, we multiply the probabilities: (4/52) × (3/51) = 12/2652, which simplifies to 1/221. Therefore, the correct probability that both drawn cards are aces is 1/221.

Option A:
Option A (1/13) usually comes from considering only the probability that a single card is an ace (4/52 = 1/13) and ignoring that we need two consecutive aces. It does not incorporate the second dependent draw, so it does not represent the joint probability of both events occurring. Because it treats the problem as if only one ace were required, it significantly overestimates the true probability. Hence, 1/13 is not correct for “both cards are aces.”

Option B:
Option B (1/169) typically arises from incorrectly assuming that the two draws are independent with replacement and computing (1/13) × (1/13). However, in this question the cards are drawn without replacement, which means the second probability changes after the first card is drawn. Using 1/169 ignores the reduced deck on the second draw and does not reflect the correct dependent-event structure. Therefore, it underestimates the correct probability.

Option C:
Option C (2/221) looks close to the correct answer but is exactly double 1/221, and there is no step in the correct derivation that would justify multiplying the final probability by 2. This kind of error often happens when students miscount favourable outcomes or mistakenly think order does not matter when they have already accounted for it. Since the exact product (4/52) × (3/51) simplifies to 1/221, any multiple of that value, including 2/221, is incorrect.

Option D:
Option D (1/221) matches the precise result from multiplying the probability of the first ace and the second ace without replacement: (4/52) × (3/51). The numerator 12 counts the ordered ways to pick two aces in succession, and the denominator 2652 counts all ordered pairs of cards, which simplifies to 1/221. This value correctly reflects both the limited number of aces and the shrinking deck size after the first draw. Hence, it is the only option fully consistent with the rules of probability for dependent events.

A is correct because conditional probability measures A given B. B is the definition formula P(A|B) = P(A ∩ B)/P(B) (when P(B) > 0). C is true because Bayes’ theorem is used to update probabilities of hypotheses/causes when evidence is known. D is true for independent events since B does not change the probability of A. E is false because P(A|B) and P(not A|B) are complementary conditional probabilities and must sum to 1, not exceed 1. Therefore, A, B, C and D only are correct.

Option A:
Option A is incomplete because it omits D, which is a key property of independent events (conditioning does not change probability).

Option B:
Option B is correct because it includes all true statements (A, B, C, D) and excludes E, which wrongly claims the sum of complementary conditional probabilities is greater than 1.

Option C:
Option C is incorrect because it includes E (false) and also omits A and D, both of which are true statements about conditional probability and independence.

Option D:
Option D is incorrect because it includes E (false) and excludes B and C, which are fundamental definitions/uses of conditional probability and Bayes’ theorem.