A parck of playing cards was found to contain only 51 cards. If the first 13 card which are examined are all black, If P is the probobility that the missed one is red. Then the value of 3P is ___________

Answer

To solve the problem, we need to find the probability \( P \) that the missing card from a pack of 51 cards is red, given that the first 13 cards examined are all black. We will then calculate \( 3P \).

### Step-by-Step Solution:

1. **Understand the Problem**:
- A standard deck has 52 cards: 26 black (clubs and spades) and 26 red (hearts and diamonds).
- We have a pack of 51 cards, and the first 13 cards examined are all black.
- We need to find the probability that the missing card is red.

2. **Define Events**:
- Let \( A \) be the event that the missing card is red.
- Let \( B \) be the event that the missing card is black.
- Let \( C \) be the event that the first 13 cards examined are all black.

3. **Calculate Probabilities**:
- We need to find \( P(A | C) \), the probability that the missing card is red given that the first 13 cards are black.

4. **Use Bayes' Theorem**:
\[ P(A | C) = \frac{P(C | A) \cdot P(A)}{P(C)} \]

5. **Calculate Each Probability**:
- **Prior Probability \( P(A) \)**: The probability that the missing card is red:
\[ P(A) = \frac{26}{52} = \frac{1}{2} \]
- **Prior Probability \( P(B) \)**: The probability that the missing card is black:
\[ P(B) = \frac{26}{52} = \frac{1}{2} \]

- **Calculate \( P(C | A) \)**: If the missing card is red, there are 26 black cards left. The probability of drawing 13 black cards from 26 black cards:
\[ P(C | A) = \frac{26}{51} \cdot \frac{25}{50} \cdot \frac{24}{49} \cdots \cdot \frac{14}{39} \]
This can be simplified as:
\[ P(C | A) = \frac{26!}{(26-13)!} \cdot \frac{13!}{51 \cdot 50 \cdots 39} \]

- **Calculate \( P(C | B) \)**: If the missing card is black, there are 25 black cards left. The probability of drawing 13 black cards from 25 black cards:
\[ P(C | B) = \frac{25}{51} \cdot \frac{24}{50} \cdots \cdot \frac{13}{39} \]

6. **Calculate \( P(C) \)**:
\[ P(C) = P(C | A) \cdot P(A) + P(C | B) \cdot P(B) \]

7. **Substituting Values**:
- Substitute the values into Bayes' theorem to find \( P(A | C) \).

8. **Final Calculation**:
After calculating \( P(A | C) \), we find \( 3P \):
\[ 3P = 3 \cdot P(A | C) \]

### Final Answer:
After performing the calculations, we find that \( 3P = 2 \).