Multiple choice

Three cards are drawn from an ordinary deck of 52 cards. Each card is replaced in the deck before the next card is drawn. What is the probability that at least one of the cards will be a spade?

  1. 3/52

  2. 9/64

  3. 3/8

  4. 37/64

  5. 3/4
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D Correct answer
Explanation

Probability of not drawing a spade in one draw is 39/52 = 3/4. For three draws with replacement, the probability of drawing no spades is (3/4)^3 = 27/64. The probability of at least one spade is 1 - 27/64 = 37/64.

AI explanation

Since the cards are replaced, this is a problem of independent events, and it is easiest to find the probability of the complementary event first. The probability of not drawing a spade on one draw is 39/52, which simplifies to 3/4. The probability of not drawing a spade in three draws is 3/4 multiplied by itself three times, which equals 27/64. Therefore, the probability of drawing at least one spade is 1 minus 27/64, which equals 37/64.

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