Three men were walking down a street talking when they met a fourth man. If the fourth man knew that two of the men always lied and the third always told the truth and he asked the three men a question, what is the probability that he got a truthful answer when one man answered?
Pr= 1/3
To calculate the probability, we need to consider all the possible scenarios. Let's break down the possibilities step by step:
Step 1: Determine who is the truthful person and who are the liars:
Since the fourth man already knows that two of them always lie and one always tells the truth, we can conclude that:
- The man who answered is either the truth-teller or one of the liars.
- The other two men who remained silent are the other liar and the other truth-teller.
Step 2: Consider the possibilities based on who answered the question:
a) Truth-teller answered:
In this case, the fourth man received a truthful answer, since the truth-teller always tells the truth.
Probability = 1/3 (since there are three possible men who could have answered).
b) Liar answered:
In this case, the fourth man received a false answer, since liars always lie.
Probability = 2/3 (since there are two possible men who could have answered).
Step 3: Calculate the probability that the truthful answer was received:
Since there are two possible scenarios (truth-teller answered or liar answered), we need to find the weighted average probability considering both cases:
Probability = (Probability of the truth-teller answering * Probability of receiving the truth) + (Probability of a liar answering * Probability of receiving a lie)
Probability = (1/3 * 1) + (2/3 * 0)
Probability = 1/3
Therefore, the probability that the fourth man received a truthful answer when one man answered is 1/3.