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

Question

Accepted Answer

To solve this problem, we need to determine the probability that the die actually shows a six given that the man reports it as a six. This is a classic example of conditional probability, which can be solved using Bayes' theorem.

Let's define the events:
- Let $ T $ be the event that the man tells the truth.
- Let $ S $ be the event that the die shows a six.
- Let $ R $ be the event that the man reports a six.

We are given:
- $ P(T) = \frac{3}{4} $, the probability that the man tells the truth.
- $ P(S) = \frac{1}{6} $, the probability that the die shows a six.
- $ P(R|T) = 1 $, the probability that the man reports a six given that he tells the truth and the die shows a six.
- $ P(R|\neg T) = \frac{1}{6} $, the probability that the man reports a six given that he lies (since he could report any number from 1 to 6).

We need to find $ P(S|R) $, the probability that the die shows a six given that the man reports a six. Using Bayes' theorem:

$$
P(S|R) = \frac{P(R|S) \cdot P(S)}{P(R)}
$$

First, calculate $ P(R) $, the total probability that the man reports a six:

$$
P(R) = P(R|T) \cdot P(T) \cdot P(S) + P(R|\neg T) \cdot P(\neg T) \cdot P(S)
$$

$$
P(R) = 1 \cdot \frac{3}{4} \cdot \frac{1}{6} + \frac{1}{6} \cdot \frac{1}{4} \cdot \frac{5}{6}
$$

$$
P(R) = \frac{3}{24} + \frac{5}{144}
$$

$$
P(R) = \frac{18}{144} + \frac{5}{144} = \frac{23}{144}
$$

Now, calculate $ P(S|R) $:

$$
P(S|R) = \frac{1 \cdot \frac{3}{4} \cdot \frac{1}{6}}{\frac{23}{144}}
$$

$$
P(S|R) = \frac{\frac{3}{24}}{\frac{23}{144}}
$$

$$
P(S|R) = \frac{3 \times 144}{24 \times 23}
$$

$$
P(S|R) = \frac{432}{552}
$$

$$
P(S|R) = \frac{3}{8}
$$

Thus, the probability that the die actually shows a six given that the man reports a six is $\frac{3}{8}$, which corresponds to Option 3.

**Explanation for Other Options:**

1. **Option 1: $\frac{1}{8}$** - This option is incorrect because it underestimates the probability. The calculation using Bayes' theorem shows that the probability is higher than $\frac{1}{8}$.

2. **Option 2: $\frac{2}{8}$** - This option is incorrect because it does not accurately reflect the probability derived from the given conditions and Bayes' theorem.

3. **Option 4: $\frac{4}{8}$** - This option is incorrect because it overestimates the probability. The correct calculation shows that the probability is $\frac{3}{8}$, not $\frac{4}{8}$.

Therefore, the correct answer is Option 3: $\frac{3}{8}$.