In my (reasoned) opinion, only the aprioristic concept of probability produces a paradox here.

Experiments cannot be paradoxical. Therefore only one answer should be correct from an a posteriori POW, ie, actually conducting the experiment and checking the right box.

What's your take on this?

Edited April 23Apr 23 by joigus
minor addition

1 hour ago, joigus said:

What's your take on this?

Paradoxes, probably don't exist...

25 minutes ago, dimreepr said:

Paradoxes, probably don't exist...

If by that you mean paradoxes exist only in our reasoning minds, and the world of facts always get past them, I think you're probably right.

Edited April 23Apr 23 by joigus
I removed a statement. Let's leave it at that.

Is the paradox not inherent in the question ?

Out of the four choices, one has to be correct so that would make it 25%.
But two out of four choices are 25%, which means 50%.

Is this one of those 'liar who claims to be lying' situations that fries android's brains in old Star Trek episodes ?

It seems to me it depends on the question One obviously expects there to be only one answer and so the question setter has a duty to make it so. But, for example, how would an examiner mark the following question?

What is the square root of 36?

a) -6

b)-3

c) +3

d) +6

5 hours ago, MigL said:

Is the paradox not inherent in the question ?

Out of the four choices, one has to be correct so that would make it 25%.
But two out of four choices are 25%, which means 50%.

Is this one of those 'liar who claims to be lying' situations that fries android's brains in old Star Trek episodes ?

It has a flavour of it. Yes. I think the paradox is always inherent in the question, by the way.

You just missed the last step. In order for the probability to be 50%, 25% must be wrong. Therefore the probability must be 0%. But don't forget the answers must be chosen at random, so no answer can really be right, but 25%, but then it's 50%, but then... An so on. Similar to Epimenides' paradox, yes. It has a looping structure.

4 hours ago, OldTony said:

It seems to me it depends on the question One obviously expects there to be only one answer and so the question setter has a duty to make it so. But, for example, how would an examiner mark the following question?

What is the square root of 36?

a) -6

b)-3

c) +3

d) +6

This one is easy, and not paradoxical, AFAICT. The right answers are a) and d)

Mathematicians do not suffer paradoxes, I think. They know the problem must be ill-posed in the first place, and so set out to identify the illness. Engineers, OTOH, are too practical. It's more for the likes of physicists to break a sweat on these things.

By the way, the words 'at random' are a potential minefield, as the don't mean anything in and of themselves. If this discussion proves to be any popular, we might end up discussing that.

Edited April 23Apr 23 by joigus
minor correction

This is a version of the Monty Hall or Newcomb problem.

If you choose A, B, C or D at random,

There are no a priori probabilities.

These can only be assigned if the chooser has no knowledge of any values assigned to A, B , C or D.

You can't conbine Bayes probabilities and Pearson probalities.

9 hours ago, joigus said:

In my (reasoned) opinion, only the aprioristic concept of probability produces a paradox here.

Experiments cannot be paradoxical. Therefore only one answer should be correct from an a posteriori POW, ie, actually conducting the experiment and checking the right box.

What's your take on this?

It's a Bayesian belief trick, since there are four choices of letter but only three choices of p value. A random choice of letter will, with many repetitions, approach 25% for each letter. But the probability for a specific p value selection approaches 50% for one value, and 25% for the other two values. So the act of random choice of letter is different from the act of random choice of p value. In essence, the paradox lies in a single overt act actually being one that's operating on two levels. It's a Bayesian trick, because there are two incompatible beliefs as to what is being picked. Choosing an answer is not the same as choosing a letter.

Edited April 23Apr 23 by TheVat
Fixed

10 minutes ago, TheVat said:

It's a Bayesian belief trick, since there are four choices of letter but only three choices of p value. A random choice of letter will, with many repetitions, approach 25% for each letter. But the probability for a specific p value selection approaches 50% for one value, and 25% for the other two values. So the act of random choice of letter is different from the act of random choice of p value. In essence, the paradox lies in a single overt act actually being one that's operating on two levels. It's a Bayesian trick, because there are two incompatible beliefs as to what is being picked. Choosing an answer is not the same as choosing a letter.

It is only Bayesian if a priori probabilities can be assigned.

This cannot be done with Pearsonian probabilities.

One other piece of information not supplied to the chooser of random A, B, C or D is that the assigned probabilities must add up to 1.
Pearson has no such requirement as all he is assigning is the probability that A, B, C, or D is 'correct' without any knowledge of the 'right answer', or even if any one of them is 'correct'.

A better way to analyse this is to consider the probability of being wrong, not right

26 minutes ago, studiot said:

There are no a priori probabilities.

These can only be assigned if the chooser has no knowledge of any values assigned to A, B , C or D.

10 minutes ago, TheVat said:

So the act of random choice of letter is different from the random choice of p value.

I think the essence of both these comments is pretty similar. Yes, knowledge, or even ballparking, intention, etc, of the person guessing essentially changes the probability distribution.

I have developed the habit to actually ask, 'what do you mean "random"? According to what probability distribution?' Most people get confused, but I think I know what I'm asking.

The moment you know something, or venture to guess something, or think you know something, the probability distribution of your answers already changes.

1 minute ago, joigus said:

The moment you know something, or venture to guess something, or think you know something, the probability distribution of your answers already changes.

Yes agreed.

But from what you said, that isn't the case here.

The chooser is not being asked to assign probabilities, she is being asked to pick A, B, C or D at random full stop.

And remember that any random variable can have any value between 0 and 1.

Edited April 23Apr 23 by studiot

2 minutes ago, studiot said:

The chooser is not being asked to assign probabilities, she is being asked to pick A, B, C or D at random full stop.

Well, that's why @TheVat 's comment comes in handy too. If 'at random' means something like 'throw a die' = 'just pick one' (Laplacian probability), it would be 1/4 each.

If it means 'take a guess based on intuition' there's another probability distribution.

If it means 'give the problem to machine X' there's another, and if given to machine Y, still another.

If it's: 'ask many people and calculate the probability á la Pearson', then probably (sorry for the pun) it would be zero, as a continuous distribution in (0,1) assigns zero probability to any particular value (and, as you know, zero probability doesn't imply impossibility).

Etc.

I think the paradox is a well-thought-out one and it illustrates a problem of using the concept of probability in too-loose a way.

20 minutes ago, joigus said:

Well, that's why @TheVat 's comment comes in handy too. If 'at random' means something like 'throw a die' = 'just pick one' (Laplacian probability), it would be 1/4 each.

If it means 'take a guess based on intuition' there's another probability distribution.

If it means 'give the problem to machine X' there's another, and if given to machine Y, still another.

If it's: 'ask many people and calculate the probability á la Pearson', then probably (sorry for the pun) it would be zero, as a continuous distribution in (0,1) assigns zero probability to any particular value (and, as you know, zero probability doesn't imply impossibility).

Etc.

I think the paradox is a well-thought-out one and it illustrates a problem of using the concept of probability in too-loose a way.

But what is 'the question' .

You wrote

If you choose an answer to this question.....

But you never defined what the question was.

If you happened to choose a random answer C you could be correct as C was 0 if the assigned answer was 'none of these'.

This is (thankfully) becoming common in multiple choice questions.

It appears to me that this is not really a question about multiple choice questions.

As has been pointed out we do not know the parameters given to the question writer such as how many answers should be correct (including none).

For some reason it brings to my mind an old conundrum where you are given a card and written on both sides of it are the words "The statement on the other side of this card is not true"

3 hours ago, OldTony said:

It appears to me that this is not really a question about multiple choice questions.

As has been pointed out we do not know the parameters given to the question writer such as how many answers should be correct (including none).

For some reason it brings to my mind an old conundrum where you are given a card and written on both sides of it are the words "The statement on the other side of this card is not true"

You must have gone to the same junior school I did.

+1

Although we also passed round a piece of pape ( we couldn't afford card) with PTO writtten on both sides.

This leads to the same infinite loop more clearly than the truth value as it is easier to demonstrate that PTO has no truth value, but it is also correct for the statement you referred to.

3 hours ago, studiot said:

You must have gone to the same junior school I did.

+1

Although we also passed round a piece of pape ( we couldn't afford card) with PTO writtten on both sides.

This leads to the same infinite loop more clearly than the truth value as it is easier to demonstrate that PTO has no truth value, but it is also correct for the statement you referred to.

Lol - I just looked up where you live - we didn't go to the same school but I spent a total of 11 years out my total of 22 years service at RAF Locking, near Weston-Super-Mare as an instructor and also a trade standards examiner. In the latter role I spent 4 years working on all aspects of multiple choice tests. More importantly I met and married a Somerset lass and we recently celebrated our 67th anniversary.

16 minutes ago, OldTony said:

Lol - I just looked up where you live - we didn't go to the same school but I spent a total of 11 years out my total of 22 years service at RAF Locking, near Weston-Super-Mare as an instructor and also a trade standards examiner. In the latter role I spent 4 years working on all aspects of multiple choice tests. More importantly I met and married a Somerset lass and we recently celebrated our 67th anniversary.

I think I can probably get closer than Weston, although I can thoroughly recommend a visit to the helicopter museum, since I went to Loughborough University in the sixties.
There were a lot of students from Marconi in those days.

The way I see it, the most offending term in the posing of the problem is the mischievous use of the expression 'at random'.

Whether we adopt a frequentist approach or an aprioristic one, distributions such us, eg,

A: 31.7 % B: 19.2 % C: 16.1 % D: 33 %

and,

A: 12.5 % B: 23.9 % C: 11.2 % D: 52.4 %

are exactly as 'at random' as each other.

Too often, people confuse 'perfectly random behaviour' with 'equal probabilities' (Laplace).

The best cautionary tale I know about this is Bertrand's paradox:

https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

On 4/23/2026 at 1:27 PM, joigus said:

In my (reasoned) opinion, only the aprioristic concept of probability produces a paradox here.

Experiments cannot be paradoxical. Therefore only one answer should be correct from an a posteriori POW, ie, actually conducting the experiment and checking the right box.

What's your take on this?

After a little thought, I shall define a 'random answer' as the result of one toss of a fair coin. If it is heads, the choice would be B; and if tails, C.

Ideally, B and C each have a 50% of being selected, B clearly being a uniquely correct answer for this methodology.

I do however recall being downmarked in exams for this kind of reasoning. Too 'engineeringy' perhaps. (It works; don't ask questions).

The semantic trick is that it's worded as "choose an answer ..." and not "choose a letter." So any answer immediately negates itself even (and this is the beauty part) when it is 0. You can choose randomly (coins, twirling tetrahedrons, whatev), you can choose deliberately, you will never be correct. A semantic trick is that any answer promptly invalidates itself.

If each letter was a different value, and only one was .25, then the problem would become trivial and easily resolved as .25.

11 hours ago, OldTony said:

and we recently celebrated our 67th anniversary

Congratulations, and many more ...

4 hours ago, MigL said:

Congratulations, and many more ...

Thank you, you are very kind. We are both chugging along with the help of modern science - bypass and pacemaker for me and a heart valve and pacemaker for my wife

11 hours ago, TheVat said:

The semantic trick is that it's worded as "choose an answer ..." and not "choose a letter." So any answer immediately negates itself even (and this is the beauty part) when it is 0. You can choose randomly (coins, twirling tetrahedrons, whatev), you can choose deliberately, you will never be correct. A semantic trick is that any answer promptly invalidates itself.

If each letter was a different value, and only one was .25, then the problem would become trivial and easily resolved as .25.

Yes, provided you are invited to just pick a letter. But even in that case, if you asked people to try to guess an answer that makes it correct, the sheer fact that people wanted to get it right would shift the probability distribution to more than .25 chance, only because they think that's the right answer! I think that's what @studiot meant when he compared this to the Monty Hall problem. I don't think it's a pure Monty Hall problem with conditional probability hidden in some semantic way. I think there's also a more-or-less hidden self-reference here that contributes to the paradox.

14 hours ago, sethoflagos said:

After a little thought, I shall define a 'random answer' as the result of one toss of a fair coin.

Yes, but then you're assuming a probability distribution: .25, .25, .25, .25 and you cannot escape the paradox either. .25+.25 = .5 so .25 would no longer be 'correct'. It is designed in such a way that, if you want to get it right, you'll get it wrong.

17 minutes ago, joigus said:

Yes, but then you're assuming a probability distribution: .25, .25, .25, .25 and you cannot escape the paradox either.

I don't follow your logic. Surely I'm enforcing a probability distribution of 0,50,50,0 in order to subvert its ill-posed nature.

I say 'ill-posed' on two counts: firstly, it is posing two different questions at the same time, the conventional understanding of which contradict each other; and secondly, there is no 'correct' choice offered for the 'non-random' answer. It's basically asking you to select the capital of Albania from four types of jellyfish.

PS... Perhaps that last bit was hyperbolic: it isn't a category error as such.

This is just an observation but seems relevant to the discussion. It is very difficult, apparently, for the human mind to produce a long list of truly random numbers. There are recognised tests for randomness and rather strangely, I think, that even if you produced a list of numbers by a method such as rolling a dice it may well produce a list of numbers that would fail a test for randomness. For that reason you can purchase a book of random numbers that agree with the rules of randomness.