Three cards from a pack of 52 cards are lost. One card is drawn from the remaining cards. If drawn card is heart, find the probability that the lost cards were all hearts

Answer

To solve the problem, we will use Bayes' theorem. We need to find the probability that all three lost cards were hearts given that the drawn card is a heart.

Let's denote:
- Event A: All lost cards are hearts.
- Event B: The drawn card is a heart.

According to Bayes' theorem, we have:

\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]

### Step 1: Calculate \( P(B|A) \)

If all three lost cards are hearts, then the number of hearts remaining in the deck is:

\[ 13 - 3 = 10 \]

The total number of cards remaining in the deck is:

\[ 52 - 3 = 49 \]

Thus, the probability that the drawn card is a heart given that all lost cards are hearts is:

\[ P(B|A) = \frac{10}{49} \]

### Step 2: Calculate \( P(A) \)

Now we need to find the probability that all three lost cards are hearts. The probability of drawing three hearts in succession without replacement is calculated as follows:

1. The probability of the first card being a heart:
\[ P(\text{1st card is heart}) = \frac{13}{52} \]

2. The probability of the second card being a heart (after one heart has been drawn):
\[ P(\text{2nd card is heart}) = \frac{12}{51} \]

3. The probability of the third card being a heart (after two hearts have been drawn):
\[ P(\text{3rd card is heart}) = \frac{11}{50} \]

Thus, the total probability \( P(A) \) is:

\[ P(A) = \frac{13}{52} \cdot \frac{12}{51} \cdot \frac{11}{50} \]

Calculating this gives:

\[ P(A) = \frac{13 \cdot 12 \cdot 11}{52 \cdot 51 \cdot 50} = \frac{1716}{132600} = \frac{11}{825} \]

### Step 3: Calculate \( P(B) \)

Next, we need to find \( P(B) \), the probability that the drawn card is a heart. There are 13 hearts in a full deck of 52 cards, so:

\[ P(B) = \frac{13}{52} = \frac{1}{4} \]

### Step 4: Substitute into Bayes' theorem

Now we can substitute these values into Bayes' theorem:

\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{\left(\frac{10}{49}\right) \cdot \left(\frac{11}{825}\right)}{\frac{1}{4}} \]

Calculating this step-by-step:

1. Calculate the numerator:
\[ \frac{10}{49} \cdot \frac{11}{825} = \frac{110}{40425} \]

2. Now divide by \( P(B) \):
\[ P(A|B) = \frac{\frac{110}{40425}}{\frac{1}{4}} = \frac{110}{40425} \cdot 4 = \frac{440}{40425} \]

3. Simplifying \( \frac{440}{40425} \) gives us approximately \( 0.0109 \).

### Final Answer

Thus, the probability that all lost cards were hearts given that the drawn card is a heart is approximately:

\[ P(A|B) \approx 0.0109 \text{ or } 1.09\% \]