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    Question

    A deck of 52 cards is used. Two king cards and one spade are removed from the deck. Then, two cards are drawn randomly one after the other without replacement. What is the probability that both cards drawn are spades?

    Statements I:  The two removed kings are of red suits (hearts and diamonds).

    Statements II:  After removing the spade card, 11 spades remain in the deck.

    A Statement I alone is sufficient but statement II alone is not sufficient. Correct Answer Incorrect Answer
    B Statement II alone is sufficient but statement I alone is not sufficient. Correct Answer Incorrect Answer
    C Both statements together are sufficient, but neither statement alone is sufficient. Correct Answer Incorrect Answer
    D Each statement alone is sufficient. Correct Answer Incorrect Answer
    E Statements I and II together are not sufficient. Correct Answer Incorrect Answer

    Solution

    Statement I: Knowing that the removed kings are hearts and diamonds does not give direct information about the spades left in the deck, making it insufficient on its own. Statement II: Knowing that 11 spades remain is crucial, as we can calculate the probability of drawing two spades one after the other without replacement:  Combining both statements: Even though statement I doesn’t directly contribute to finding the probability, statement II alone is sufficient since it provides the exact count of spades remaining. Answer: B. Statement II alone is sufficient but statement I alone is not sufficient.

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