A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be heart then the probability of the missing card to be a heart is:

$\frac{11}{50}$

The correct answer is Option (4) → $\frac{11}{50}$ **

Total hearts in a full deck = 13

One card is missing. Let:

$H$ = event that the missing card is a heart

$E$ = event that the two drawn cards (from remaining 51) are both hearts

We need $P(H\mid E)$.

Using Bayes’ theorem:

$P(H\mid E)=\frac{P(E\mid H)\,P(H)}{P(E\mid H)\,P(H)+P(E\mid H^{c})\,P(H^{c})}$

$P(H)=\frac{13}{52}=\frac{1}{4},\quad P(H^{c})=\frac{3}{4}$

If the missing card is a heart, remaining hearts = 12, remaining cards = 51:

$P(E\mid H)=\frac{12}{51}\cdot\frac{11}{50}$

If the missing card is not a heart, remaining hearts = 13:

$P(E\mid H^{c})=\frac{13}{51}\cdot\frac{12}{50}$

Substitute:

$P(H\mid E)= \frac{\frac{12}{51}\cdot\frac{11}{50}\cdot\frac{1}{4}} {\frac{12}{51}\cdot\frac{11}{50}\cdot\frac{1}{4} +\frac{13}{51}\cdot\frac{12}{50}\cdot\frac{3}{4}}$

Cancel common factors:

$=\frac{11} {11+39}=\frac{11}{50}$

The required probability is $\frac{11}{50}$.