In this probabilistic paradox, The question seems apparently simple:
By chance by randomly an answer to this questionwhat is the chance that it is correct?
The possible answers are those reported in the figure: 25%; 0%; 50%; 25%. What will the right one? Let’s start thinking together.
We know that if in a question with 4 distinct answers And only one is correct, then the chance to spot the right one at random would be 1 out of 4that is, of the 25%. So the correct answer may seem 25%. However, if 25% were actually the correct answer, we immediately notice that in our question there are Two responses equal to 25%and therefore two possible correct answers! But then, the probability of spotting the correct answer becomes 2 out of 4that is, the 50%. This means that the correct answer is not 25%, but 50%! But even in this case, if 50% were the correct answer, we note that 50% appear only once and therefore the probability of spotting it is 1 in 4, that is, 25%. Here we are entered a paradoxical loop: If the correct answer was 25%, the probability of spot on it would be 50%, but then the correct answer is 50%, which however has a probability of being spot on equal to 25%, which is then the correct answer … and so on!
The only plausible response that remains at this point is it 0%. But if 0% is the correct responseit means that the probability of spotting the response at random is nothing, which paraphrase means that there is no correct response! This means that 0% is be right and wrong! Another paradox.
In short, any response we try to give, places us in front of a logical paradox. But what is this type of paradox based on?
When a question refers to itself
This paradox is based on a truck typical of the paradoxes: the question that is asked is self -test, that is, he wonders about herself. The same dynamic can be found in the famous paradox of liar, In which a person exclaims “I’m lying”, and this simple phrase leads to a logical paradox because it is impossible to establish whether those who are speaking is sincere or not.
In this case, the question asks to establish one chance though depends on the nature of the request same, thus leading to a logical paradox because we are not able to establish a priori what are the events we are considering.
Let’s try to understand it by reformulating the question itself. If instead of the original question we asked:
By changing an answer at random to a questionwhat is the probability that it is correct if we know that One response is correct?
In this case, the answer is definitely 25% Although in our box there are two solutions equal to 25%. In fact, now we are in front of a pure probabilistic questionin which we are asked, in general, what is the probability of success if you have 4 possible events available, of which only one is correct. The probabilistic answer to this question is 25%, and it doesn’t matter if in our 4 options there are two correct (because they are identical), because it is not our question that has to respect the condition of having only one correct answer. In short, in this question, the question does not refer to itself!
The same is true if we explicit that we are talking about a generic question – not our own question! – in which of the 4 possible answers, 2 are correct. From a probabilistic point of view we certainly know that the correct response And 50%because what is asking us for the question is – in general – what the probability of success is when you have 4 possible events available to which 2 lead to victory. In this case, the question is purely probable and speaks to us of a very specific case: it is not telling us that our same question has two possible answers, but is talking about the probabilistic situation in which 2 responses out of 4 are correct.
