Two cards are drawn from a deck of cards in which one card is missing. It is known that the drawn cards are both club cards. Find the probability that the missing card is a club card.
- \(\frac{1}4\)
- \(\frac{3}{4}\)
- \(\frac{11}{50}\)
- \(\frac{13}{50}\)
- \(\frac{9}{50}\)
The Correct Option is C
Solution and Explanation
There are two possibilities, the missing card is a club card or the missing card is not a club card. The probability that the missing card is a club card is ¼, and the probability that it is not a club card is ¾.
Case I: When the missing card is a club card:
The probability that the two cards drawn are club cards = \(\frac{12C_2}{51C_2} = \frac{12\times11}{51\times50}\)
Case II: When the missing card is not a club card:
The probability that the two cards drawn are club cards = \(\frac{13C_2}{51C_2} = \frac{13\times12}{51\times50}\)
By Baye’s Rule,
The required probability = \(\frac{\frac{1}{4}\times\frac{12\times11}{51\times50} }{ \frac{1}{4}\times\frac{12\times11}{51\times50}+\frac{3}{4}\times\frac{13\times12}{51\times50 }}= \frac{11}{50}\)
Hence, option C is the correct option.
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