## A Probabilistic Paradox

### Tanya Khovanova and Alexey Radul

We all heard this paradoxical statement:

This statement is false.

Or a variation:

True or False: The correct answer to this question is ‘False’.

Recently we received a link to the following puzzle, which is similar to the statement above, but has a cute probabilistic twist:

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

- 25%
- 50%
- 60%
- 25%

There are four answers, so you can choose a given answer with probability 25%. But oops, this answer appears twice. Is the correct answer 50%? No, it is not, because there is only one answer 50%. You can see that none of the answers are correct, hence, the answer to the question—the chance to be correct—is 0. Now is the time to introduce our new puzzle:

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

- 25%
- 50%
- 0%
- 25%

## Bernat:

http://500motivators.com/plog-content/thumbs/motivate/me/large/44-oh-i-see-what-you-did-there.jpg

19 November 2011, 7:06 pm## Tim Anderson:

If we choose the answer with a weighted random variable weighted according to the percentage answer, the correct answer is 50%

20 November 2011, 6:50 pm## Animesh:

I don’t think that makes a difference. Still none of the answers are correct.

22 November 2011, 7:25 am## Alexis:

“If you choose an answer to this question” is not a question…You have to actually ask a question.

22 November 2011, 12:01 pm## Phil:

Obviously, the question is “What is the chance you will be correct(If you choose an answer to this question at random)?”

22 November 2011, 9:17 pm## Jake:

There are 4 possible answers- with the percentage of that number being chosen at 25%

23 November 2011, 6:00 pmSince 2 answers are the same it would then increase your oddds to 50% which appears only once which is 25% time. Therefore if you chose 25% or 50% you would be wrong(wrong 75%). That leaves 0% but since there is a chance (25%) you would chose 0 (which would be correct)0 is wrong. With this being said you would be wrong 100% of the time. If you are always wrong you have 0% chance of choosing correctly which appears 25%, and 25% apperas 2x so you have 50% chance of getting that so 100% you would correct and since there is not 100% you would have 0% chance of getting right which is 25% which occurs 50% of the time. The answer is you have 0% chance of getting it right because if you get it right than 0 is not correct.

## Ryan:

the first puzzle assumes that one of the answers is correct. there is no place for assumptions here since this puzzle must be solved using logic. any number of those answers could be correct. assuming that there is in fact one correct answer: the question is asking me what the chances are that ill be correct in choosing a random answer, not what the chances are that the random answer i choose will be correct. if that had been the case, then the question would have asked: “If you choose an answer to this question at random, what is the chance that answer will be correct”.

23 November 2011, 10:11 pm## Animesh:

In my understanding here’s how I interpret it…..

If I choose the answer A) which is 25%, then what is the chance I will be correct + (extra statement) “25% of the time”….and so on for other options…

Coz then there are two right answers out of 4…

if I choose 50% then what is the chance I will be correct 50% of the time…..there is one right answer…so I will be correct 25% of the time.

If I choose 60%…what is the chance I will be correct 60% o the time….—same reason—

But if I choose none of the answers , what is the chance I will be correct 0% of the time…..hence 0%.

Again in the next one if there is a zero…then chance I will be correct 0% is again contradictory….so no answer….

Won’t it be interesting if multiple choices are allowed….??

24 November 2011, 1:12 pm## him:

All that is is a multiple choice question without the proper answer as an option

24 November 2011, 5:28 pm## Mu:

Fifty percent because my answer will either be false or correct.

24 November 2011, 11:31 pm## LongyAUS:

I’ve said it before and I’ll say it again. The answer is B. You can only be correct or incorrect, so any answer randomly chosen has a 50% chance of being correct.

25 November 2011, 5:08 amIt’s a simple question with a simple answer that people try and complicate.

## ObsessiveMathsFreak:

This is ultimately a self referential paradox and cannot be answered definitely in any mathematical framework we now possess. The answer depends upon the answer.

25 November 2011, 5:15 pm## Rob:

@Alexis: The question is “What is the chance of being correct?” It is a valid question.

@longy: But there is only a 25% chance of choosing 50% at random.

25 November 2011, 6:09 pm## X:

So I made 2 pages of math, and then i throw them away. This is what I think:

8 March 2012, 3:21 amSo I must chose an answer to the question “what is the chance to be correct…?”

1. I want to see the chance to be correct.

1a. If I chose A, I chose D too. So the chance it`s not 25%, it`s 50%. So B. But 50%, it`s only 1/4 in my answer, so it`s not correct. = A, B, D ~ paradox, or a chance X

1b. If I chose B, it`s probably correct, no? I don`t know. So let`s say the chance it`s Y.

1c. We have X+Y chance to be correct.

2. If I want to see the chance to be incorrect.

2a. If I chose A, i chose D, so A, D = incorrect, I have 50% chance of beeing incorrect, but B is not the answer. So, again A, B, D ~ paradox, let`s say Z

2b. If I chose C, the chance is a the same number Y

BUT WAIT!

paradox X it`s the same with paradox Z!

Paradox X + number Y = Paradox Z + number Y!

We talk about chance, so it`s a proportion, and the chance of beeing correct + the chance of beeing incorrect are complementary. The chances are equal, so we have 50% chance of beeing correct and 50% chance of beeing incorrect.

So the answer is B, 50%.

(but I`s sure of that 50%. So the chances are 25% :)) )

## X:

I read the qiestion again. And again, and again… and:

“If you choose an answer to this question at random”

CHOSE ANY ANSWER, a, b, c or d.

and now…

“what is the chance you will be correct?”

in my mind… “well, I chose A”.

the chances of beeing correct are 50%!

You have to calculate the chances of beeing correct of a chosen answer!

8 March 2012, 3:38 amB. 50%, now i`m sure 100% 🙂

## moliate:

I would choose either A or D

If I assume A or D is correct I would have 25% chance of randomly selecting B; which in turn is the probability that I selected A or D.

31 July 2012, 11:49 am## jd:

“If you choose an answer to this question at random, what is the chance you will be correct?”

Nowhere does it indicate you must choose from the four choices listed below, nor are there any instructions that say you must even consider those options. It is superflous information, yet everyone assumes you MUST treat the question as such because they have been culturally trained to.

The answer to this logic question is 50%, because if you choose any answer to any question at random, you are either correct or incorrect. You might even be someone who believes that questions have no answers and that existence has no meaning (0%) OR you are a relativist, who believes questions can have more than one answer, and that there is no right or wrong, black or white, but everything is gray (100%) But even in those cases, it is human nature to think that the people who are diametrically opposed to *your* point-of-view are “wrong” and that your view is “correct” …

So, again: 50%.

8 January 2013, 7:26 pm## Gavin:

moliate: By choosing A you would need to choose D as well, which bring you to B (50%), but you never chose B, which bring me to jd: if you chose B you would have a 25% chance, so B is also incorrect as it references A and / or D. The Answer is ‘infinite’ or a sideways ‘8’ in probable statistics.

It’s the same as: “This statement is false”. Is the statement True or False?

The statement is true, the answer of the statement is false, therefore the statement is true and false or neither true nor false…

So by selecting 1, 2, 3, all or none of the answers would be correct and incorrect or neither correct nor incorrect.

11 November 2013, 8:56 am## Sanjeevi:

It’s very close to a paradox but not . The answer is 1/2.

Its becomes a paradox only if you think the answer that you are supposed to give is listed in the options.

Before proceeding into answer, lets understand the question. The idea behind the question is to find the probability of getting a correct answer , with one option listed twice.

So, now change the options with below identical ones.

a) RED

b)BLUE

c:) GREEN

d.)RED.

Now , we don’t know whether RED or BLUE or GREEN is correct. Hence, the chance of correct answer becomes 2/4 = 1/2.

It only leads to misconception only if you mess up with options listed in the question as your answer’s options , which is never said so in the question.

13 March 2014, 8:50 am