# Uncertainty Puzzle

Here’s a puzzle from understanding uncertainty:

“Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? A) 25% B) 50% C) 60% D) 25%”

This suggests a variant:

“Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? A’) 25% B’) 50% C’) 0% D’) 25%”

My answers are ….

0% for the first, ‘none’ for the second. These aren’t in the list of options supplied, but (at least for British school exams) this is not a novelty. (It would be  a ‘paradox’ , as claimed in the original source, if there were some law of logic that said that one of the answers provided is true: but there isn’t. It is only normally a mistake.)

My reasoning is …

For the first is to consider A…D in turn and to show that each is wrong:

• If A were correct, then so would D be, and hence the probability of the correct answer being selected is 50%. Thus  A and D are both wrong.
• If B were correct, then the probability of the correct answer being selected is 25%. Thus B is wrong. Similarly for C.

For the second, I claim that the answer is ‘None’. Again I consider each possible answer in turn and show that each is wrong.

• If C’ is correct then the probability of the correct answer being selected is 25%. Thus C’ is wrong.
• Since the answers are all wrong, if the probability of the correct answer being selected is a number, then it is 0%, and so C’ must be true. But C’ is not true, so the required probability is not a number.

To put it another way, ‘chance’ in the question is meaningless since the structure of the question mimics Russell’s paradox. This puzzle seems more appealing than Keynes’ examples of non-numeric probabilities, which assume a richness of life experiences.

## See Also

Similar puzzles here. My probability theory notes may help.

Dave Marsay

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

### 23 Responses to Uncertainty Puzzle

1. Marco says:

The problem I see in the first variant is that is a multiple choice answer. Therefore you do not have an option to circle that says 0%. Therefore even for the first part it has no answer.

• Dave Marsay says:

Agreed. It is one of those annoying multiple-choice questions for which none of the answers is correct. It was presented as a paradox, but I do not think that it is. But what about my variant?

• Curt Welch says:

Silly goose, the answer is “none” for both. This isn’t a hard puzzle, nor is the answer disputed. The answer is none. There is no answer. It’s exactly the same type of math “puzzle” as this one: Solve for x: x = x + 1. There is no finite real value of x that makes this true, just as there is no answer in the multiple choice question that makes the question true. It’s very easy in math to ask a question which has no answer, and this multiple choice question is just one of an infinite number of such questions that can be asked in math that have no answer.

• thefarmerman says:

It is not a paradox, it is a common logical fallacy called ambiguity.

• Dave Marsay says:

Possibly. Which word or phrase do you regard as ambiguous? Maybe I need to disambiguate it?

2. yarivh says:

I think the answer to the variant is should be the same as for the original. If we limit the possible answers to the four options, then both are unanswerable (“None”); if we allow the unlisted answer of 0% (let’s call it E) for the original then we should allow it (E’) for the variant.
In the variant, then, C’ and E’ are both 0%, but only E’ is correct – precisely BECAUSE it cannot be randomly selected, i.e. the chance of its random selection is 0%, unlike for C’.

• Dave Marsay says:

Thanks. I guess it depends what you think the rules are for multiple-choice questions. The key difference, to me, is that in the first case the question has a numeric answer, it just isn’t among the options listed. This should never happen, but it sometimes does, e.g. due to misprunts. In the second case, I argue that no number is the answer to the question. Thus the two ‘none’s are different: the first is ‘none of the options provided’, the second ‘no number at all’. My variant invokes Russell’s paradox and Keynes and Knight’s notions of non-numeric uncertainty. The original doesn’t, or doesn’t so directly.

I am less clear about the solution to some of the similar puzzles. Ideas?

• yarivh says:

Yeah, darn those misprunts! 😉

If I’m understanding correctly, your statement that the two ‘none’s are different is based on the fact that the “actual” answer of 0% is available as an option in the variant, but not in the original, therefore in the variant we cannot say that the answer is actually 0%.
My reasoning is that we CAN say that the actual answer to the variant is 0%, using exactly the same logic as in the original. Whichever option we choose would still not be correct, irrespective of the fact that C’ is 0%. The fact that choosing C’ affects the answer means that choosing C’ is qualitatively different to simply stating that the answer is 0% (rather than “C’, 0%”). So I’d say the fact that the actual answer is 0% does not contradict the statement that choosing the answer “0%” from the four options does not result in having chosen the correct answer.
What do you think on that point?

I haven’t looked at the other puzzles yet, but I will!

• Damon says:

I think many people are confused…this is NOT a multiple choice question exactly…read it and you’ll see the question is essentially asking for a probability. It is a question about a [multiple choice question]. e.g. Is the following question appropriate for 4th grade: “what is the derivative of sin x? a) 2x, b) -cos x, c) tan x, d) cos x ?” The answer is “NO”. Hopefully this shows that a question about a multiple choice question need not be answered by a letter.

In this case the answer, if it is a well formed question HAVING an answer is a probability. So “0” is a legitimate form for an answer (I am not saying it is right or wrong) but “none” doesn’t quite make sense unless by ‘none’ you mean ‘0 probability or 0% chance/probability”

• Damon says:

I’ve just noticed a change you made that unintionally may have alterned this puzzle…you put ‘multiple choice’ in front….I’ve not seen that before. My comment above applies to the puzzle without those words. Then the puzzle is clearly asking for a numerical probability answer. THen my comment above applies and the silly stuff about letter answers goes away…it’s not really part of the problem…the problem is self reference etc.

3. Dave Marsay says:

yarivh. My counter-argument is as follows.

I assume that in both cases each option has at least 25% chance of being chosen. (E.g. ‘25%’ has a 50% chance of being chosen.) Anything not on the list has a 0% chance.

In the original case, if you claim that the answer is 0% then as ‘0%’ is not on the list of options it has 0% chance of being chosen. Correct.

In the variant, we seem to agree that saying ‘C” would be wrong, since it has a 25% chance of being picked. But if you say ‘Not C’ but 0%’ I will say that C’ IS 0%, and hence that you are wrong. So maybe it is down to who interprets the rules.

Here is another variant:

“Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? A) 25% B) 50% C) 60% D) 26%”

I would accept either ‘A’ or ‘25%’, you – I think – would insist on ‘A’ or ‘A 25%’. But then I think ‘A’ would be a different answer to ‘D’ in the original and you have a perfectly good puzzle, but a different one to the one that I had intended.

• yarivh says:

In general, I agree with you that it’s a question of interpretation and that the disagreement comes down to whether giving the answer “0%” necessarily means giving the answer “C'”. My contention that it doesn’t is based on the original question, where we interpret “choose an answer” to mean one of the four options (A, B, C, D), rather than one of the three values (25%, 50%, 60%). We have assigned selection probabilities of 25% each to A and D because choosing one means that we are not choosing the other.
However, I concede that it CAN be interpreted differently.

I didn’t quite get the meaning of your last sentence – I have always considered that A and D are different answers (with the same value), but that doesn’t seem to yield a valid solution… in fact my reasoning is identical to yours for this question. (The crucial difference is that A and D are identical, while C’ and my E’ are not, because their selection probabilities are different.)

I had a look at the other puzzles, by the way, but they seem to be far more open-ended than this one, so I didn’t find them as interesting!

• Dave Marsay says:

We seem to think about multiple-choice questions differently, and perhaps I should have dealt with these issues in the original post. For some wording, there is a problem, but there was more uncertainty than I had intended. Sorry.

The other puzzles are also intended to illustrate some uncertainty. It would be good to be able to re-word them to make this clearer, but then one does need to understand what people make of them.

4. MicaelCabral says:

Hi, i think the answer is c) 60% ’cause we have 5 probabilities to choose the right answer (a) b) c) d) or nothing). If we say that 0% is a right answer and 25% is a good answer too, we have 3/5 prob to choose the right one. So we have 60%! (:

5. Lucas Moya says:

Well I disagree, because if we say that the chance is 0% we are saying that this is impossible, but if we have 4 possible answers: A)X B)Y C)a D)X and we pick one at random we can say that if we choose A or D the chances of being right are the 50% and if we pick B or C the chances will be 25%.
I’m really sorry for my english and my maths, I’m not really good but y wanted to try 🙂

6. Gabriel Young says:

Dear , the key to the riddle is ” an answer to this question at random”
No reason out the answer , guess. If there are 2 possible outcomes guess , right or wrong , so the correct answer to the riddle is B

• Dave Marsay says:

You appear to be applying the principal of indifference. But there are two issues with it. (1) It only applies where the problem is totally symmetric between the alternatives, which is not the case here. (2) If you choose at random then you only have a 25% chance of choosing B, yet you are claiming a 50% chance.

7. Mihail Ghinea says:

Language is a form of communicating information. Language is not information.

A question is a request of information. When a question requests the information found within itself you end up with what is called a loop, a paradox or what I like to call “a severe case of the incorrect use of language”.

That being said, if we “play along”, while the answer to the primary puzzle is obviously 0% because we can prove that all given choises are wrong one by one, I do have to congratulate you for your variant puzzle, you managed to close the loop.

My answer to your variant, since I cannot pick “0” or “none”, would be “undetermined”.

Please pardon my english, not my primary language.

8. nono says:

The paradox of the first question is in theory resolvable if we postulate an answer key that lists the correct answer as A and only A or D and only D; in other words, the implicit question becomes ‘one of the letters A-D is the pre-chosen correct answer, what is the probability of choosing the correct letter at random?’

• Dave Marsay says:

Thanks. You have raised in my mind a question about ‘correctness’. Is the correct one the one that the person setting the question has nominated, or is it the one (or ones) that meet some ‘objective’ standard? I think maybe I have often assumed the latter when the former might have been more expedient. 😉