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# Take a guess

## Question

If you chose to answer this question completely at random, what is the probability you will be correct?

1. 25%
2. 50%
3. 0%
4. 25%

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Spoiler

But really, it should be fairly simple, I think:

Each option has a 25% chance of being randomly picked. That is the correct answer: 25%.

BUT there are two options where "25%" is the answer. This means that the answer that you pick has a 50% chance of being "25%", which is STILL the correct answer regardless of the probability of picking an answer that HAS that value. So the probability that you are correct is 50% which is answer 2...

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This is a fun one...

Spoiler

I'd say the probability of randomly choosing the correct answer is 0%. If any of answers (1, 2, 3, and/or 4) is considered correct, it results in a contradiction. So, none of the answers may be considered correct (including option 3).

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42 minutes ago, ThunderCloud said:

This is a fun one...

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I'd say the probability of randomly choosing the correct answer is 0%. If any of answers (1, 2, 3, and/or 4) is considered correct, it results in a contradiction. So, none of the answers may be considered correct (including option 3).

Then again...

Spoiler

It also seems contradictory to claim that the probability is 0% and yet answer choice #3 is not correct. So, perhaps the only consistent assessment is that the probability of guessing a correct answer is undefined. The question is, after all, a paradox.

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15 hours ago, Pickett said:

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But really, it should be fairly simple, I think:

Each option has a 25% chance of being randomly picked. That is the correct answer: 25%.

BUT there are two options where "25%" is the answer. This means that the answer that you pick has a 50% chance of being "25%", which is STILL the correct answer regardless of the probability of picking an answer that HAS that value. So the probability that you are correct is 50% which is answer 2...

So...

The probability that you choose the correct value of 25% is 50% but since 50% is not 25%, you'd be part of the 50% that guessed wrong?

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Spoiler

By definition, p(to be correct at random)=(number of correct answers)/(number of answers)

As the (number of correct answers) is not known, p cannot be calculated.

Even if the possible answers were {15%, 25%, 35%, 45%}, the problem would remain meaningless. One could believe 25% to be the correct answer. Why should it be correct? Don't tell me

It is correct because it is correct.

Edited by harey
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Spoiler

If the question is to be interpreted to suggest that one of (1), (2), (3), or (4) is the correct answer, and that one of those four answers is chosen randomly such that each has a 25% probability of selection, then it can be shown that the question has no answer:

Among the answer choices given, there are three distinct probabilities: 50%, 25%, and 0%. If the correct answer is 50%, then since this is only one out of four answer choices, the probability must in fact be 25%. That is, by assuming the correct answer is 50%, we obtain that it is not 50%. Similarly, if we assume the correct answer is 25%, then since two out of four answer choices are correct, the real answer must be 50%… i.e., not 25%, a contradiction once again. If we assume that the answer is 0%, then answer (3) is correct and the probability of choosing it is 25%, not 0%, once again a contradiction. Finally, if we assume that the probability is anything else not listed as an answer choice, then the probability of choosing it is 0% (and therefore, 25%,  etc…), meaning it is, in fact, listed among the answer choices. Once again, a contradiction. For any 0% ≤ p ≤ 100%, if we assume the correct answer is p, then we must conclude that it is not p. There cannot be a consistent answer.

On the other hand, if the question is interpreted to suggest that it is answered as an entirely random probability (any real number between 0% and 100%, without regard to the listed answer choices), then the probability of choosing the precisely correct answer is 0%, as non-zero probabilities can only occur over an interval (e.g., probability of choosing the correct answer ± 1%).

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