Final answer:
To find the probability that the die is actually a six when the man reports a six, we can use Bayes' theorem.
By calculating the individual probabilities and plugging them into the formula, we find that the probability is 3/7.
Step-by-step explanation:
Let's denote the event of the man reporting a six as event A, and the event of the actual roll resulting in a six as event B.
We know that the probability of speaking a lie is 1/3 and the probability of rolling a six on a fair die is 1/6.
We can use Bayes' theorem to calculate the probability:
P(B|A) = (P(A|B) * P(B)) / P(A)
P(A|B) is the probability of reporting a six given that the actual roll was a six, which is 1.
P(B) is the probability of the actual roll resulting in a six, which is 1/6.
P(A) is the probability of reporting a six, which can be calculated as:
P(A) = P(A|B) * P(B) + P(A|B') * P(B')
= 1 * 1/6 + 1/3 * 5/6
= 7/18.
Plugging these values into the Bayes' theorem formula:
P(B|A) = (1 * 1/6) / (7/18)
= 3/7.
Therefore, the probability that the actual roll is a six given that the man reports a six is 3/7.