The Lemon Fool

Apologies if this question is really simple for everyone on this board, but I'm struggling to work it out.

There is a 'Who Wants to be a Millionaire?' style-question on social media that is worded as follows:

If you chose an answer to this question at random, what is the chance that you will be correct?

A: 25%
B: 0%
C: 50%
D: 25%

Now, I have eliminated 'B' as you have to choose an answer so you have some chance of success. As there are two answers at 25% (A & C), does this mean that the correct 'chance' figure to the questions is 'C' (50%)? Or, is it one of the 25% options as there are always 4 possibilities (ignoring that two have the same answer. My best guess is 'C'.

Thanks for any help! Illogically, OLTB.

Well you have a one in four chance of being correct it would seem. It seems obvious to me that the unbiased answer has to be 25%, but I am not sure what the question is.

Dod

OLTB wrote:Apologies if this question is really simple for everyone on this board, but I'm struggling to work it out.

There is a 'Who Wants to be a Millionaire?' style-question on social media that is worded as follows:

If you chose an answer to this question at random, what is the chance that you will be correct?

A: 25%
B: 0%
C: 50%
D: 25%

Now, I have eliminated 'B' as you have to choose an answer so you have some chance of success. As there are two answers at 25% (A & C), does this mean that the correct 'chance' figure to the questions is 'C' (50%)? Or, is it one of the 25% options as there are always 4 possibilities (ignoring that two have the same answer. My best guess is 'C'.

Thanks for any help! Illogically, OLTB.

Spoiler...

It's pretty clearly intended to be a cousin of the Barber Paradox (look it up on Wikipedia if you don't know what it is) - i.e. something posing as a puzzle which contradicts itself no matter what answer you try to make solve it consistently. Specifically:

* If one tries to make an assumption that the answer is 0% consistent, then one of the four choices A-D is the right answer, so picking an answer at random has a 25% chance of being correct.
* If one tries to make an assumption that the answer is 25% consistent, then two of the four choices A-D are the right answer, so picking an answer at random has a 50% chance of being correct.
* If one tries to make an assumption that the answer is 50% consistent, then one of the four choices A-D is the right answer, so picking an answer at random has a 25% chance of being correct.
* If one tries to make an assumption that the answer is anything other than 0%, 25% or 50% consistent, then none of the four choices A-D is the right answer, so picking an answer at random has a 0% chance of being correct.

None of those situations makes the answer consistent with what it's assumed to be, as in the Barber Paradox. So basically, it's a paradox rather than a puzzle.

Or at least, it would be if the person setting it hadn't made one of the most common errors in dealing with probability puzzles. To fix the resulting flaw in the paradox, it would have had to say (with the fix emboldened):

"If you chose an answer to this question at random with equal probabilities, what is the chance that you will be correct?

A: 25%
B: 0%
C: 50%
D: 25%"

As it stands, I can for instance produce a consistent answer by deciding to select my answer by tossing a fair die and choosing A if a 1 is rolled, B if a 2 is rolled, C if a 3, 4 or 5 is rolled, and D if a 6 is rolled. If I do that, A, B and D are provably incorrect answers, as each of them has a 1/6th = 16.666...% chance of being selected at random in my chosen way, which is not what they say it is. But C has a 50% chance of being selected at random in that way, which is what it says the answer is, so it is at least a consistent answer - and it's the only consistent answer, which in some sense makes it 'correct'. But by choosing other methods to make my random choice, I can make other methods 'correct' in the same sense...

So my final verdict on it is that it's a somewhat flawed attempt at a paradox, with the flaw making it instead a puzzle that is flawed because it has multiple 'correct' answers...

Gengulphus

Thank you Gengulphus - I have looked up the paradox you mention and appreciate your reply.

Best wishes, OLTB.

There was me thinking a play on Russell's paradox.

I have a very distant[1] recollection of reading - possibly in something like Martin Gardner - of Russell's paradox and of imitators that don't quite work. It had a name that sounds much like Russell's paradox: I had "Richard's Paradox" in mind, but googling finds there is such a thing and I don't think it's what I remember.

Does that horribly vague account ring a bell with anyone here?

[1] I was probably in my early teens at the time.

Spoiler...

The probability is to some extent a distraction. This has the same result:

How many of these answers A-D are correct?
A: Only one of them
B: None of them
C: All of them
D: Only one of them

If no answers are correct, B is correct, which means that it's not true that no answers are correct.
If one answer is correct, both A and D are correct, so it's not true that only one of them is correct
If two or three of them are correct, none of the answers are correct as "two" or "three" aren't included in the possible answers.
If all four answers are correct, C is the only correct answer, so it's not true that all of them are correct.
There are no other possibilities. However you answer this question, you will not be correct.

An even shorter version, more related to the Liar Paradox - "Is the answer to this question 'no'?"
If you say no, then the answer is yes. If you say yes, then the answer is no. You're wrong either way.

How about if you choose one answer, I reveal one of the others that has a donkey behind it, and I give you the option to change?


Julian F. G. W.

(Aided and abetted by a couple of beers).

If we have a computer, and use the "computer says so, so it must be" approach, there can only be one correct answer*.
Of course we don't know which of the 25% ones the computer has been told is the right answer. But the computer will blindly reject the letter associated with the other 25% value. And there's no appeal.

* Unless I've misunderstood the WWtbaM format (I don't watch it), and sometimes they do have two answers, either of which they will accept as correct.

- Rover

Rover110 wrote:If we have a computer, and use the "computer says so, so it must be" approach, there can only be one correct answer*.
Of course we don't know which of the 25% ones the computer has been told is the right answer. But the computer will blindly reject the letter associated with the other 25% value. And there's no appeal.

* Unless I've misunderstood the WWtbaM format (I don't watch it), and sometimes they do have two answers, either of which they will accept as correct.

I watched it quite a lot in the Chris Tarrant era (which Wikipedia tells me ran from 1998 to 2014), but haven't seen even a single episode in the more recent (2018 onwards) Jeremy Clarkson era. So my knowledge of the format is all pretty old and potentially out-of-date. I'd be reasonably certain that in the episodes I watched, they never had two answers they would accept as correct, since the answers offered seemed pretty clearly incompatible with each other. But there were plenty of questions for which we were only told that the answer the contestant had given was correct, or that it was wrong and another was correct, leaving the issue of whether the other three or two answers would have been accepted unresolved - so that cannot be complete certainty.

However, I'm completely certain that WWtbaM never offered two identical answers in episodes I watched it, and just about as certain that it never did so in episodes I didn't watch either because that would have hit the headlines if it had happened. So it seems quite clear to me that this 'puzzle' does not follow the WWtbaM format, much as it tries to give the impression that it does!

Gengulphus

The basic problem here is that people assume that any string of words makes sense and can be interpreted in a meaningful and consistent way.

That most strings of words are complete nonsense should be obvious. What is the difference between a duck? One of it's legs is both the same.

Russell et. al. showed that many strings of words that apparently make sense do not, they cannot be interpreted consistently.

If you are doing math you do not assume consistency, you try to prove it. Any contradiction in a system completely invalidates the entire system. Unfortunately most other areas of human endeavour, particularly legal systems and game shows, do not recognize this and turn themselves into knots to try to accept contradictions. But since the results cannot be based on reason they are based entirely on whim.