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A box contains 8 cards. Each card has a single positive integer printed on it, and different cards may show the same integer. Two cards are then drawn at random, without replacement. What is the probability that both cards drawn show an even integer?
(1) The product of the integers on all 8 cards is an odd number.
(2) Each of the 8 integers shown is a prime number greater than 2.
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Correct answer
D
The asked probability is invariant once you know whether any even cards exist; the specific integers never matter.
Statement (1): a product of integers is odd only if every factor is odd, so all 8 cards are odd, meaning zero even cards, so the probability of drawing two even cards is 0: sufficient. The non-obvious step is reading the odd-product condition as a parity constraint, not as magnitude arithmetic. Statement (2): every prime greater than 2 is odd, so all 8 cards are odd, again zero even cards and probability 0: sufficient. Each statement alone forces the same answer by a different parity argument, so neither needs the other.
The trap (E) is the reflex that unknown labels mean an unknown probability; the discipline is that the asked quantity is invariant across every labeling each statement admits. Answer: (D).
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