At a restaurant downtown, Mr. Red, Mr. Blue, Mr. Green, Mr. White and Mr. Yellow meet for lunch. Under their coats they are wearing either a red, blue, or white shirt. They are all wearing different colored shirts from their names- so that- Mr. Red is not wearing a red shirt, Mr. Blue is not wearing a blue shirt and so on.
Each of them are likely to speak truthfully with a probability of 2/3, while the chance that exactly one of them speaks falsely is 1/3.
The man wearing a red shirt says, "Mr. Red is not wearing a yellow shirt."
The man wearing a yellow shirt says, "Mr. Yellow is wearing a red shirt."
Given that exactly one of the above statement is true, determine:
(a) The probability that Mr. Green is wearing a blue shirt.
(b) The probability that Mr. Blue is wearing a white shirt.
Since we're actually told one of the statements is true and the other false, it doesn't matter what the probabilities of those events were; they happened.
We know that Mr. Red is indeed not wearing a yellow shirt, as only red, blue and white shirts were worn, so the man in the red shirt was the one telling the truth, and the man in the yellow shirt was lying, so Mr. Yellow definitely is not wearing a red shirt. That's all we know beyond what the first paragraph says.
(a) Having nothing to go by, each of the colors red, blue and white are equally likely to the extent of our knowledge. The probability of blue is 1/3.
(b) From the first paragraph we know that Mr. Blue is not wearing blue. Only red and white are possible, and we have no reason to favor the probability of either one; any biases might favor either one, but what they favor could be equally likely to be either one. The probability is 1/2.
Edited on October 18, 2016, 10:59 am
Posted by Charlie
on 2016-10-18 10:58:05