This should make your head explode

[MarginOfError]

I'm posting this everywhere because...well, I love it. Enjoy

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%

(was published on ANZSTAT mailing list a couple of days ago).

[Jennarator]

That would depend on the answer. Since there are two 25%.

[MarginOfError]

That would depend on the answer. Since there are two 25%.

It doesn't depend on the answer; there is only one correct answer. But what is the probability of randomly selecting the correct answer?

[JudoMinja]

It looks like a paradox- as with a four answer multiple choice question the probability of randomly choosing the correct answer would be 25%. However, 25% is provided as an answer twice, making it two out of four answers or 50%. So one would think that 50% ( B ) is the correct answer, but if that is the correct answer then it is really back to being one out of four, which should be 25%.

[MarginOfError]

It looks like a paradox- as with a four answer multiple choice question the probability of randomly choosing the correct answer would be 25%. However, 25% is provided as an answer twice, making it two out of four answers or 50%. So one would think that 50% ( B ) is the correct answer, but if that is the correct answer then it is really back to being one out of four, which should be 25%.

[skippy740]

The option of 100% is missing as well... because if you KNOW the answer, you have a 100% chance of being correct. :)

[MarginOfError]

The option of 100% is missing as well... because if you KNOW the answer, you have a 100% chance of being correct. :)

But then you're not answering the question, which stipulates selecting an answer at random.

[estradling75]

It looks like a paradox- as with a four answer multiple choice question the probability of randomly choosing the correct answer would be 25%. However, 25% is provided as an answer twice, making it two out of four answers or 50%. So one would think that 50% ( B ) is the correct answer, but if that is the correct answer then it is really back to being one out of four, which should be 25%.

Take it farther... Because of this the chance of guessing the correct answer is 0 which is not a option

[Guest xforeverxmetalx]

It looks like a paradox- as with a four answer multiple choice question the probability of randomly choosing the correct answer would be 25%. However, 25% is provided as an answer twice, making it two out of four answers or 50%. So one would think that 50% ( B ) is the correct answer, but if that is the correct answer then it is really back to being one out of four, which should be 25%.

That was basically my train of thought when I first saw it elsewhere.

Is there actually a correct answer, or is it just something someone made up to confuse people?

[skippy740]

But then you're not answering the question, which stipulates selecting an answer at random.

I like to think outside the box.

[JudoMinja]

Take it farther... Because of this the chance of guessing the correct answer is 0 which is not a option

Hmm... okay. So we have a multiple choice question where the correct answer is not one of the options to choose from. Selecting an answer at random will never bring up the correct result, so the probability of choosing correctly is zero. It's like, if someone has a bag of yellow and red pencils and asks you the probability of pulling out a blue pencil at random. There are no blue pencils inside the bag, so the probability is zero.

[MarginOfError]

Hmm... okay. So we have a multiple choice question where the correct answer is not one of the options to choose from. Selecting an answer at random will never bring up the correct result, so the probability of choosing correctly is zero. It's like, if someone has a bag of yellow and red pencils and asks you the probability of pulling out a blue pencil at random. There are no blue pencils inside the bag, so the probability is zero.

It isn't like that at all, however. You could randomly select the correct answer, but the correct answer cannot be selected with the probability of the correct answer. More directly, the probability of selecting the correct answer is not equal to the value given in the correct answer. Thus, the answer and the probability contradict each other and the universe ceases to exist.

[JudoMinja]

It isn't like that at all, however. You could randomly select the correct answer, but the correct answer cannot be selected with the probability of the correct answer. More directly, the probability of selecting the correct answer is not equal to the value given in the correct answer. Thus, the answer and the probability contradict each other and the universe ceases to exist.

So you're saying that the answers provided are not answers to the "probability" question at all? Just answers? And that one of the four answers is the "correct" one?

[Vort]

It looks like a paradox- as with a four answer multiple choice question the probability of randomly choosing the correct answer would be 25%. However, 25% is provided as an answer twice, making it two out of four answers or 50%. So one would think that 50% ( B ) is the correct answer, but if that is the correct answer then it is really back to being one out of four, which should be 25%.

At the risk of being pedantic, it's a silly and nonsensical question, and not because it's self-referential. It's poorly worded. It's like saying, "If pigs had wings, what is the probability that today is Wednesday?" "The correct answer" is an undefined quantity, so any non-negative percentage up to 100% might be "the correct answer".

The question obviously wants to be self-referential. If we assume that (1) the correct answer is given in the choices and (2) that the correct answer is cleverly self-referential, then the correct answer is unobtainable. It must be 25% -- the odds of selecting one of four -- but since there are two 25% answers, it is 50%. But of course, there is only one 50% answer, so it cannot be that.

It strives to be exactly the same silliness as doing this sort of thing:

When I answer a multiple choice question, I always pick:

a) any answer but the first.

b) the last answer, in all cases.

c) I do not pick any answers in multiple choice questions.

I must admit that I appreciate such humor, when it is pulled off correctly.

[james12]

In order to think about this question with any hope of coming to an understanding I think the following assumptions have to be made. (These are not too farfetched based on the fact that it is a multiple choice test.)

1. Outcomes are collectively exhaustive: One of the four possibilities must be correct.

2. Each answer is mutually exclusive: If one answer is correct the other three are not.

3. Each answer is independent: One outcome does not affect the next outcome.

With those under our belt I would look at the random aspect. This is where the results are not clear. I may select an option of A, B, C, or D at random but in most multiple choice tests the placement of the correct answer is not random. Meaning that most of the time the individual determining the location of the correct answer is selective. If this is the case then at least the relative percentages between A, B, C and D appear plausible. In other words, a teacher will most likely place correct answers under options C then B.

[RescueMom]

42.

[Guest mormonmusic]

The formula for a probability is:

# Events in the event space/# events in the sample space.

In other words, number of right choices/number of possible choices.

In this case there is one event -- the right answer. The sample space is up for discusson because there are four choices. But two are the same. For me, a) and d) are really the same choice. So, I would say the probability of getting the answer right if you select at random is

1/3 or 33.33%

My approach collapses a) and d) into one choice since they are the same value. This makes the events in the sample space equal to 3, the denominator in my formula above.

[MarginOfError]

The formula for a probability is:

# Events in the event space/# events in the sample space.

In other words, number of right choices/number of possible choices.

In this case there is one event -- the right answer. The sample space is up for discusson because there are four choices. But two are the same. For me, a) and d) are really the same choice. So, I would say the probability of getting the answer right if you select at random is

1/3 or 33.33%

My approach collapses a) and d) into one choice since they are the same value. This makes the events in the sample space equal to 3, the denominator in my formula above.

That isn't really correct. A) and D) are distinct and separate events that happen to share the same value. You could say that this is more like a permutation than a combination.

[sister_in_faith]

I'm sorry, what was the question again?

[prisonchaplain]

I'm posting this everywhere because...well, I love it. Enjoy

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%

(was published on ANZSTAT mailing list a couple of days ago).

The correct answer should be "not enough information given." In order to choose an answer to "this question" we would need to know what "this question" is. Even though the answer is stipulated to be random, "this question" cannot be reasonably answered unless we know what it is. Therefore the question is invalid, and I nominate it as a write in candidate for ... oops! I'm a mod, I can't do that here...uh...NEVER MIND!

[prisonchaplain]

I'm sorry, what was the question again?

My point exactly!

[Dravin]

I'm posting this everywhere because...well, I love it. Enjoy

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%

(was published on ANZSTAT mailing list a couple of days ago).

If we assume the correct and possible answers are in terms of A, B, C, or D, think a multiple choice test, and only one of the four is correct then the chance one will randomly choose correctly is 25% (number desired outcomes/number of total outcomes). Now if we aren't thinking a multiple choice test and assuming there is a correct answer you have 1/∞* (though I suppose for all practical purpose 1/∞ and (>1)/∞ are equivalent they both end up an infinitely small number) or an infinitely small number. For those asking what the question is, if we're truly answering randomly the question doesn't matter, any response within the domain is equally likely. In the case of multiple choice the domain is implied and if an unknown open question is presented since the domain is not given to us (or discoverable by us) there is nothing to restrict our responses.

* Unless the domain is sufficiently restricted (such as with a multiple choice test, or the sides of a die) there are an infinite number of possible outcomes.

Edited November 2, 2011 by Dravin

[Guest mormonmusic]

Ok, then, what defines an event? The value, or the event label? For me, the "right answer" is the "event" you are looking for, and so it's the value that matters. Who cares if it's A or D as long as the value is correct? It's on the basis of the right answer that the moniker "correct answer" is awarded.

As a teacher of 17 years, if I had two answers and they were correct, but my scoring sheet only recognized A) and not D) for example, there would be classwide student backlash and low evaluations if I didnt' accept either A or D as the right answer.

Therefore, I maintain that the probability of a right answer is 1/3.

And by the way, you don't have to know what the right answer is to make this conclusion. It's true for all possible right answers if you consider A) and D) the same event, and only one event in the sample space.

[Dravin]

Therefore, I maintain that the probability of a right answer is 1/3.

If you maintain that the value 1/3 (or 33%) is the correct answer (as opposed to A, B, C, or D being the right answer regardless of it's value) then there is a 0% chance of randomly picking the correct answer from A, B, C, or D. Why? Because 1/3 or 33% isn't one of the options, it's akin to having this:

What year was the US Declaration of Independence adopted?

A) 1724 AD

B) 1984 AD

C) 600 BC

D) 2010 AD

If the correct answer is the value 1776 AD there is a 0% chance of it being randomly selected within the domain of A, B, C, and D.

Edited November 2, 2011 by Dravin

[MarginOfError]

Ok, then, what defines an event? The value, or the event label? For me, the "right answer" is the "event" you are looking for, and so it's the value that matters. Who cares if it's A or D as long as the value is correct? It's on the basis of the right answer that the moniker "correct answer" is awarded.

As a teacher of 17 years, if I had two answers and they were correct, but my scoring sheet only recognized A) and not D) for example, there would be classwide student backlash and low evaluations if I didnt' accept either A or D as the right answer.

Therefore, I maintain that the probability of a right answer is 1/3.

And by the way, you don't have to know what the right answer is to make this conclusion. It's true for all possible right answers if you consider A) and D) the same event, and only one event in the sample space.

An event is a possible outcome from the sample space. In this case, A, B, C, or D. The fact that A and D share the value does not make them the same event. You could represent it as the figure below.

Notice that choosing red is not the same as choosing green, even if they share a common label.

Even in the application of a multiple choice test, if you have a question

1 + 1 = ?

a) 2

b) 3

c) 4

d) 2

The probability of randomly selecting the correct answer from the three options is not .3333.

Edited November 2, 2011 by MarginOfError
Thanks Dravin