Question

A pack of playing cards was found to contain only 51 cards. If the first 13 cards, which are examined are all red, the probability that the missing card is black, is

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{5}{9}$
• D. $\frac{2}{9}$

Answer 1

The correct option is B

$\frac{2}{3}$

There are two cases:

Case 1: 38 cards - 13 red and 25 black [1 black card missing]

Case 2: 38 cards - 12 red and 26 black [1 red card missing]

The probability that the missing card is black, is $\frac{{}^{38}{C}_{25}}{{}^{38}{C}_{25}{+}^{38}{C}_{26}}$

$=\frac{\frac{38!}{25!\times 13!}}{\frac{38!}{25!\times 13!}+\frac{38!}{26!\times 12!}}$

$=\frac{2}{3}$