Question

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

• A. $\frac{5}{17}$
• B. $\frac{4}{17}$
• C. $\frac{3}{17}$
• D. $\frac{2}{17}$

Answer 1

Answer: (B)

Let $A$ and $B$ are the events that card lost is spade and card drawn is spade.
$P(A)=\frac{1}{4},P(\bar{A})=\frac{3}{4}$
$P(B/A)=\frac{12}{51},P(B/\bar{A})=\frac{13}{51}$
$P(A/B)=\frac{P(A)\cdot P(B/A)}{P(A)\cdot P(B/A)+P(\bar{A})\cdot P(B/A)}$
$=\frac{\frac{1}{4}\cdot \frac{12}{51}}{\frac{1}{4}\cdot \frac{12}{51}+\frac{3}{4}\cdot \frac{13}{51}}$
$=\frac{12}{51}=\frac{4}{17}$