Question

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.

• A. $\frac{1}4$
• B. $\frac{3}{4}$
• C. $\frac{11}{50}$
• D. $\frac{13}{50}$
• E. $\frac{9}{50}$

Answer 1

The Correct Option is C

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.