Question

A card from a deck of $52$ cards is lost. From the remaining cards, two cards are drawn with replacement and found to be clubs, then the probability that

• A. missing card is a club is $\frac{16}{523}$
• B. missing card is a spade is $\frac{13}{51}$
• C. missing card is a spade is $\frac{1}{4}$
• D. missing card is a club is $\frac{1}{4}$

Answer 1

Answer: (A), (C)

missing card is a spade is $\frac{1}{4}$

Let
$E_1:$ missing card is a club
$E_2:$ missing card is other than club
$E:$ drawn cards are club

$\therefore P\left(E_1|E\right)=\frac{P\left(E|E_1\right)P\left(E_1\right)}{P\left(E|E_1\right)P\left(E_1\right)+P\left(E|E_2\right)P\left(E_2\right)}$

$P\left(E_1|E\right)=\frac{{\left(\frac{12}{51}\right)}^2\times \frac{1}{4}}{{\left(\frac{12}{51}\right)}^2\times \frac{1}{4}+{\left(\frac{39}{51}\right)}^2\times \frac{3}{4}}=\frac{16}{523}$

Probability of lost card being spades is $\frac{1}{4}$