A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a king. Find the probability of the lost card being a king.

$\frac{3}{51}$

The correct answer is Option (3) → $\frac{3}{51}$ ##

$E_1$: Lost card is king, $E_2$: Lost card is not king.

Let $A$: a card drawn from remaining pack is king.

$P(E_1) = \frac{4}{52}, P(E_2) = \frac{48}{52}$

$P\left(\frac{A}{E_1}\right) = \frac{3}{51}, P\left(\frac{A}{E_2}\right) = \frac{4}{51}$

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

$ P\left(\frac{E_1}{A}\right)= \frac{\frac{4}{52} \cdot \frac{3}{51}}{\frac{4}{52} \cdot \frac{3}{51} + \frac{48}{52} \cdot \frac{4}{51}}$

$ P\left(\frac{E_1}{A}\right) = \frac{4\times 3}{4\times 3+ 48\times 4}$

$ P\left(\frac{E_1}{A}\right)= \frac{3}{51}$