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.a)7/12b)1/4c)1/2d)3/8Correct answer is option 'D'. Can you explain this answer?

Question

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.&#8203;a)7/12b)1/4c)1/2d)3/8Correct answer is option 'D'. Can you explain this answer?

Answer

Answer

Let E be the event that the man reports that six occurs in the throwing of the die and
let S1 be the event that six occurs and S2 be the event that six does not occur. Then P(S1) = Probability that six occurs = 1/6
P(S2) = Probability that six does not occur = 5/6
P(E|S1) = Probability that the man reports that six occurs when six has actually occurred on the die
= Probability that the man speaks the truth = 3/4
P(E|S2) = Probability that the man reports that six occurs when six has not actually occurred on the die
= Probability that the man does not speak the truth = 1 - 3/4 = 1/4
Thus, by Bayes' theorem, we get
P(S1|E) = Probability that the report of the man that six has occurred is actually a six
⇒P(S1/E) = (P(S1)P(E/S1))/(P(S1)P(E/S1) + P(S2)P(E/S2))
= (1/6 x 3/4)/(1/6 x 3/4 + 5/6 x 1/4)
= 1/8 x 24/8
= 3/8

Answer

Given:
A man speaks the truth 3 out of 4 times, which means he lies 1 out of 4 times.

To Find:
The probability that the man is telling the truth when he reports that a die roll is a six.

Solution:
Let's denote the event of the man telling the truth as T and the event of the man lying as L.

Step 1: Assign probabilities to events:
P(T) = 3/4 (probability of telling the truth)
P(L) = 1/4 (probability of lying)

Step 2: Consider the event of rolling a six:
P(6) = 1/6 (probability of rolling a six on a fair die)

Step 3: Calculate the probability that the man tells the truth when he reports a six:
We need to find P(T|6), which represents the probability of the man telling the truth given that he reports a six.

Using Bayes' theorem:
P(T|6) = (P(6|T) * P(T)) / P(6)

Step 4: Determine the remaining probabilities:
P(6|T) = 1 (the man always tells the truth when the roll is a six)

Step 5: Calculate P(6):
P(6) = P(6|T) * P(T) + P(6|L) * P(L)
= 1 * 3/4 + 1/6 * 1/4
= 3/4 + 1/24
= 18/24 + 1/24
= 19/24

Step 6: Substitute the values into Bayes' theorem:
P(T|6) = (1 * 3/4) / (19/24)
= 3/4 * 24/19
= 72/76
= 9/19

So, the probability that the man is telling the truth when he reports a six is 9/19, which is approximately 0.4737 or 47.37%.

Answer:
The correct answer is option D) 3/8.

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