The Probabilistic Liar's Paradox
Kapitoly: The Three Door Problem, The Probabilistic Liar's Paradox, The question mark paradox, Simpson's Paradox, The medical paradox, The St. Petersburg Paradox, Non-transitive cubes
This example is similar to the classic liar's paradox, only translated into probability.
Assignment
If you randomly choose an answer to this question, what is the chance that the answer given will be correct?
- A) 25 %
- B) 50 %
- C) 0 %
- D) 25 %
The original liar's paradox
The example suffers from the classic problem of roles that refer to themselves. That is why the liar's paradox was mentioned above, which can read as follows: 'This sentence is false'. The paradox is that if the sentence is indeed false, then the sentence is telling the truth. But if the sentence says the truth, it cannot be true, because it says so itself!
Solution
In the solution, we assume a uniform probability distribution, so we can choose each answer randomly with probability 25 %.
- Assume that answer A is correct. The important thing here is that there is an equal percentage in answer D, so the probability that we choose answer 25 % is 50 %. Therefore, neither answer A nor D can be correct.
- The probability that we choose answer B is 25 %. Therefore, answer B cannot be correct either, because it says that we have 50% chance of choosing the correct answer.
- The probability that we choose answer C is 25 %. Therefore, even answer C cannot be correct, because that says we have 0% chance of choosing the correct answer.
As you can see, none of the answers are correct. So what chance do we have of choosing the correct answer? If none of the answers are correct, then we 0% have a chance of choosing the correct answer. Which, of course, conflicts with the fact that we actually have a 25% chance of choosing the answer that says we have a 0% chance.
So the problem doesn't have a solution, it's kind of a nice paradox, like the classic liar's paradox.