A man is known to speak the truth 3 out of 4 times. He throws a die arid report that it is six. The probability that it actually a six is _____

To solve this problem, we can use **Bayes' Theorem**, which helps us find the probability of an event given that another event has occurred.

Let's define the events:
- \( A \): The event that the die shows a six.
- \( B \): The event that the man reports a six.

We need to find \( P(A | B) \), the probability that the die shows a six given that the man reports a six.

### Given Information:
- The probability that the die shows a six: \( P(A) = \frac{1}{6} \)
- The probability that the die does not show a six: \( P(\overline{A}) = \frac{5}{6} \)
- The probability that the man tells the truth: \( P(B | A) = \frac{3}{4} \)
- The probability that the man lies: \( P(B | \overline{A}) = \frac{1}{4} \) (since if the die is not a six, he can still report six by lying).

### Step-by-Step Calculation:

Using Bayes' Theorem:

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

Substituting the known values:

\[
P(A | B) = \frac{\frac{3}{4} \cdot \frac{1}{6}}{\frac{3}{4} \cdot \frac{1}{6} + \frac{1}{4} \cdot \frac{5}{6}}
\]

### Simplifying the expression:

- Numerator: \( \frac{3}{4} \cdot \frac{1}{6} = \frac{3}{24} = \frac{1}{8} \)
- Denominator: \( \frac{3}{4} \cdot \frac{1}{6} + \frac{1}{4} \cdot \frac{5}{6} = \frac{1}{8} + \frac{5}{24} \)

To combine the fractions in the denominator, find a common denominator:

\[
\frac{1}{8} + \frac{5}{24} = \frac{3}{24} + \frac{5}{24} = \frac{8}{24} = \frac{1}{3}
\]

Finally:

\[
P(A | B) = \frac{\frac{1}{8}}{\frac{1}{3}} = \frac{1}{8} \times \frac{3}{1} = \frac{3}{8}
\]

### Final Answer:
The probability that the die actually shows a six given that the man reports a six is \( \frac{3}{8} \).

Would you like more details on this calculation, or do you have any questions? Here are some related questions you might find useful:

1. What is Bayes' Theorem and how is it used in probability?
2. How would the result change if the man were more or less truthful?
3. What if the man reports a different number on the die?
4. How do you calculate probabilities when multiple conditions are involved?
5. What are some real-world applications of Bayes' Theorem?

**Tip:** Bayes' Theorem is a powerful tool for updating probabilities based on new information. Practice using it in various scenarios to strengthen your understanding!

Explore a probability problem involving a man's truthfulness about a dice roll using Bayes' Theorem. Understand how to calculate the probability that a dice shows a six given the man reports a six. Includes step-by-step solution and related questions.

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