A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is an even number. Find the probability that is actually even.

Answer

To solve the problem step by step, we will use Bayes' theorem to find the probability that the die actually shows an even number given that the man reports it as even.

### Step 1: Define the Events
Let:
- \( A \): The event that the die shows an even number.
- \( B \): The event that the man reports an even number.

### Step 2: Find the Probabilities
1. **Probability that the man speaks the truth**:
\[ P(\text{Truth}) = \frac{3}{4} \]
2. **Probability that the man lies**:
\[ P(\text{Lie}) = 1 - P(\text{Truth}) = 1 - \frac{3}{4} = \frac{1}{4} \]

3. **Probability that the die shows an even number**:
The even numbers on a die are 2, 4, and 6. Thus, there are 3 even numbers out of 6 total numbers.
\[ P(A) = \frac{3}{6} = \frac{1}{2} \]

4. **Probability that the die shows an odd number**:
The odd numbers on a die are 1, 3, and 5. Thus, there are also 3 odd numbers out of 6 total numbers.
\[ P(A') = \frac{3}{6} = \frac{1}{2} \]

### Step 3: Find \( P(B|A) \) and \( P(B|A') \)
1. **Probability that the man reports even given that it is even**:
If the die shows an even number, the man tells the truth with probability \( \frac{3}{4} \).
\[ P(B|A) = P(\text{Truth}) = \frac{3}{4} \]

2. **Probability that the man reports even given that it is odd**:
If the die shows an odd number, the man lies and reports it as even with probability \( \frac{1}{4} \).
\[ P(B|A') = P(\text{Lie}) = \frac{1}{4} \]

### Step 4: Use Bayes' Theorem
We want to find \( P(A|B) \):
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]

### Step 5: Calculate \( P(B) \)
Using the law of total probability:
\[ P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A') \]
Substituting the known values:
\[ P(B) = \left(\frac{3}{4} \cdot \frac{1}{2}\right) + \left(\frac{1}{4} \cdot \frac{1}{2}\right) \]
\[ P(B) = \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

### Step 6: Substitute Back to Find \( P(A|B) \)
Now substituting back into Bayes' theorem:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{\left(\frac{3}{4}\right) \cdot \left(\frac{1}{2}\right)}{\frac{1}{2}} \]
\[ P(A|B) = \frac{\frac{3}{8}}{\frac{1}{2}} = \frac{3}{8} \cdot \frac{2}{1} = \frac{3}{4} \]

### Final Answer
The probability that the die actually shows an even number given that the man reports it as even is:
\[ \boxed{\frac{3}{4}} \]