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 hearts. Find the probability of the missing card to be a heart.

Explanation

To solve the problem of finding the probability that the missing card is a heart given that two drawn cards from the remaining cards are hearts, we can follow these steps:

### Step 1: Define Events
Let:
- \( A_1 \): The event that the missing card is a heart.
- \( A_2 \): The event that the missing card is not a heart.
- \( A \): The event that two drawn cards are hearts.

### Step 2: Determine Prior Probabilities
The total number of cards in a standard deck is 52, with 13 hearts. Therefore:
- \( P(A_1) = \frac{13}{52} = \frac{1}{4} \)
- \( P(A_2) = \frac{39}{52} = \frac{3}{4} \)

### Step 3: Calculate Conditional Probabilities
Next, we need to calculate the probabilities of drawing two hearts given each case of the missing card.

1. **If the missing card is a heart (\( A_1 \))**:
- Remaining hearts = 12 (since one heart is missing)
- Remaining total cards = 51
- The probability of drawing 2 hearts:
\[ P(A | A_1) = \frac{\binom{12}{2}}{\binom{51}{2}} = \frac{66}{1275} \]

2. **If the missing card is not a heart (\( A_2 \))**:
- Remaining hearts = 13
- Remaining total cards = 51
- The probability of drawing 2 hearts:
\[ P(A | A_2) = \frac{\binom{13}{2}}{\binom{51}{2}} = \frac{78}{1275} \]

### Step 4: Use the Law of Total Probability
Now we can find \( P(A) \):
\[ P(A) = P(A_1) \cdot P(A | A_1) + P(A_2) \cdot P(A | A_2) \]
Substituting the values:
\[ P(A) = \frac{1}{4} \cdot \frac{66}{1275} + \frac{3}{4} \cdot \frac{78}{1275} \]
Calculating this gives:
\[ P(A) = \frac{66}{5100} + \frac{234}{5100} = \frac{300}{5100} = \frac{1}{17} \]

### Step 5: Apply Bayes' Theorem
We want to find \( P(A_1 | A) \):
\[ P(A_1 | A) = \frac{P(A | A_1) \cdot P(A_1)}{P(A)} \]
Substituting the known values:
\[ P(A_1 | A) = \frac{\frac{66}{1275} \cdot \frac{1}{4}}{\frac{1}{17}} = \frac{\frac{66}{5100}}{\frac{1}{17}} = \frac{66 \cdot 17}{5100} = \frac{1122}{5100} \]
Simplifying gives:
\[ P(A_1 | A) = \frac{11}{50} \]

### Final Answer
The probability that the missing card is a heart is \( \frac{11}{50} \).

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