Multiple choice

A card is lost from a pack of $52$ playing cards. From the remainder of the pack, two cards are drawn and are found to be spade. The probability that the missing card is a spade is :

  1. $ \dfrac{11}{50} $
  2. $ \dfrac{13}{50} $
  3. $ \dfrac{17}{50} $
  4. $ \dfrac{23}{50} $
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A Correct answer
Explanation

Let E be the event that two drawn cards are spades. Let M_s be the event that the missing card is a spade. P(M_s | E) = P(E | M_s) * P(M_s) / P(E). P(M_s) = 13/52 = 1/4. P(E | M_s) = (12/51) * (11/50). P(E | not M_s) = (13/51) * (12/50). Using Bayes' theorem, the result is 11/50.

AI explanation

Using Bayes' theorem, we compare the likelihood of drawing two spades from the remaining 51 cards whether the missing card is a spade or not. The probability of the missing card being a spade is 13/52, and if it is, the chance of drawing two spades is 12C2 divided by 51C2, which is 66/1275. The probability of the missing card being a non-spade is 39/52, and if it is, the chance of drawing two spades is 13C2 divided by 51C2, which is 78/1275. The combined probability of drawing two spades is (13/52 times 66/1275) plus (39/52 times 78/1275), which equals 11/85. The final probability of the missing card being a spade is the first scenario's probability divided by the total probability, (13/52 times 66/1275) divided by 11/85, which results in 11/50.

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