# A probability paradox?

I recently tweeted a link to this problem drawn on a blackboard, which got a lot of retweets.

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 is a fun question whose paradoxical, self-referential nature quickly reveals itself – A) seems to be fine until one realizes the D) option is also 25%.

A quick search reveals hundreds of discussion contributions of this problem, for example here and here and from a year ago. People often appear very confident that their answer is the only possible solution.

I am no logician and so unqualified to place this within the grand structures of mathematical paradoxes. I have not waded through all the discussions and so there may be something I have missed, but in among all the arguments there seem to be four conclusions that could be considered as 'correct'. These are my personal comments:

**1) There can be no solution, since the ambiguity of ‘correct’ makes the question ill-posed.**

It's true the question is ambiguous, but this still seems a bit of a cop-out.

**2) There is no solution. **

This seems to take this interpretation of the question.

*Which answer (or set of answers) of “p%”, is such that the statement ‘the probability of picking such an answer is p%’ is true? *

Then this appears to be a well-posed question, but there is no solution.

**3) 0%.**

Consider a different interpretation of the question.

*Is there a p%, such that the statement ‘the probability of picking an answer “p%” is p%’ is true?*

Then this appears a well-posed question and has the solution p = 0, even though this is not one of the answers. Of course if answer C) were changed to “0%” (as it is in this 2007 version of the question ), then this would also have no solution.

**4) We can produce any answer we want by changing the probability distribution for the choice.**

Why should ‘random’ mean an equally likely chance of picking the 4 answers? If we, say, assume the probabilities of choosing (A) (B) (C) (D) to be (10%, 20%, 60%, 10%) then the answer to either formulation (2) and (3) is now “60%”. But if we make the distribution (12.5%, 15%, 60%, 12.5%) then we seem to back to square one again, since there is now both a 25% chance of picking “25%”, and a 60% chance of picking “60%”.

I like conclusion 3) best, ie **0%**.

Maybe the main lesson is: ambiguity and paradox are often the basis for a good joke.

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## Comments

Mark Lewney (not verified)

Mon, 31/10/2011 - 12:45pm

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## Multi-choice problem

Tom (not verified)

Fri, 25/11/2011 - 9:35pm

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## But what if none of the

Bilf (not verified)

Thu, 29/12/2011 - 9:59am

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## Impossible

Dave Marsay

Wed, 01/02/2012 - 9:56pm

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## Probability paradox?

Dave Marsay

Wed, 01/02/2012 - 10:00pm

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## Oops

paradoxif ..., i.e. some things that linguistically seem to be probabilities aren't.Curt Welch

Sat, 08/03/2014 - 6:51pm

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## No Answer

Dave, your answer is clearly wrong. 0% is not a valid answer to a multiple choice question.

Let me give you a simple example to help you understand your error.

Multiple Choice: If you flip a fair coin, what is the probability of getting heads?

A) 75%

B) 25%

C) 100%

D) All of the above

E) None of the above

The answer is E. That's how multiple choice questions are answered. If you write 50% below the answers on the test paper, you would be marked wrong. Writing a number is a NOT a valid answer to a multiple choice question even if the question is "what is the probability".

If you still think that the answer to the paradox is 0%, then you have shown a basic failure in your ability to follow instructions.

Let me give you another example:

Multiple Choice: What is 1+1?

A) 22

B) 11

C) 0

What is the answer? There is no correct answer! Would this question make you start making up nonsense answers like you did in the Paradox question? Or would you just admit the question is stupid and has no answer? Why is it that people can't grasp that the paradox question simply has no valid answer? What is it about the question that causes people keep making up nonsense answers and try to argue in support of the nonsense?

It's like me saying the answer to this last question is A), and giving the reason that it is because the teacher just accidently made a typo when typing up the question and obviously meant to type a single 2! The question as written is clear, and the question as written, simply has no valid answer. End of story. Nothing else to discuss.

Blitz

Wed, 09/05/2012 - 3:23am

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## 100 %

efoula

Thu, 10/05/2012 - 9:08am

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## Basically, none of the

Blitz

Sun, 13/05/2012 - 6:14pm

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## Focus on the ?

I love this question because it had so many layers. First thought was of course, what they said in school. "If you don't know the answer, you have a one in four chance if you guess." But then you see the other 25%... OK now it's 50%... but wait now there is a is a 0%... but that's not one of the answers ether. Take the answers out, and answer THIS question.

Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? A B C D

or what if the answers were A)Blue B)Yellow C)Red D)Blue

This answer implies that there is another question. But the question states "this question"

So if I choose Yellow, Red, or ether Blue, I'm right... What if the implied question was whats my favorite color?

garybird

Thu, 17/05/2012 - 3:49pm

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## Zero

pedroAbreu

Mon, 15/10/2012 - 3:14pm

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## You always have an answer to

You always have an answer to the question :

1. How many answers are right from a set of four possible answers?

It is either 0, 1, 2, 3 or 4.

You also always have an answer to the question :

2. What is the probability of randomly picking the right answer from a set of four alternatives, given a fixed number of right answers?

It is either 0%, 25%, 50%, 75%, or 100%.

In this case they’re sort of asking you both questions simultaneously, since the content of the possible answers should be fixing and allowing you to discover both

a. the number of right answers

and

b. the right answer

Once you give an answer to either 1 or 2 you immediately get an answer to the other. In this case none of those answers is actually fixed, you should discover both at the same time: hence the (apparent) problem.

The trouble is that there are very there are some (actually very strict) constrains regarding the possible relation between a and b that are not being respected by the given possible answers: hence the incoherence.

Notice that for someone to be able to pick an answer correctly the following conditions must be met:

I) There is one correct answer ←→ There is one and only one option = 25%

II) There are two correct answers ←→ There should be two and only two options = 50%

III) There are three correct answers ←→ There should be three and only three options = 75%

IV) There are four correct answers ←→ There should be four and only four options = 100%

once you don’t respect this conditions you have an incompatibility between picking the right answer and picking the number of right answers, and so there is no way to answer the question and you should not be bogged by the appearance that there should be such an answer.

andyf

Thu, 29/08/2013 - 8:59am

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## Head... hurts...

Here's where I got to...

Suppose one of the three percentages is correct. Then:

P(correct|right(25)) = 1/2

P(correct|right(50)) = 1/4

P(correct|right(65)) = 1/4

P(correct & right(25)) = 1/2 * 1/3 = 1/6

P(correct & right(50)) = 1/4 * 1/3 = 1/12

P(correct & right(65)) = 1/4 * 1/3 = 1/12

So P(correct) = 4/12 = 1/3

None of the a,b,c,d

Try again:

Four possible answers: a, b, c, d

Correct answer just 1/4

Again: But if we look at the content of the answers, then two of them are 1/4, so it's 1/2

mariahcareyhero1993

Fri, 07/03/2014 - 6:41pm

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## but..........................................!!!!!!!!!!!!!!!!!!!

hole34

Sun, 01/06/2014 - 10:28am

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## The answer -----------------