Best Statistics Question Ever

Several readers have sent this one in1. One Raymond Johnson (or was it Ryan Grover?) said he found the following picture on this blackboard (I pulled my copy from Flowing Data):

Best statistics question ever

Try to think of an answer before reading further.

Recall that all statements of logic, which includes probability statements, are conditional on stated evidence. As a consequence of this, not all statements have an exact truth value, and so not all have a probability.

Example: What is truth/probability of this statement:

    S = “This sentence is false” ?

If S is true (has probability 1), then it is true that “this sentence is false,” therefore the sentence is not true. But if S is false (has probability 0), then it is false that “this sentence is false,” therefore it is true. The circle is complete!

Self-referentiality in math and logic is always dangerous, but it has led to profound insights. Gödel anyone? Is self-referentiality present in the best statistics question ever (BSQE)? Before we come to that, let’s examine this multiple-choice question:

  • Who wrote the play Hamlet?
    1. A: William Shakespeare
    2. B: PG Wodehouse
    3. C: John Milton
    4. D: William Shakespeare

The correct answer is A or D. Further, there is no universal rule that the right answer has to appear only once, though I believe all would agree that it should appear at least once. We therefore assume that the right answer appears at least once in the BSQE.

Finally, Jaynes has shown us (in his magisterial Probability Theory: The Logic of Science) that if any ambiguity in the wording exists, in particular around the phrase “at random”, then the same statement can have two or more different probabilities depending on how the wording is interpreted (example).

Be wary, because any or all three of these complications might apply to the BSQE.

The English phrase, “If you choose an answer to this question at random” might mean, “If you were to chose an answer to this question at random,” or it could mean, “You must chose an answer to this question at random.” If the latter, then the BSQE is self-defeating because requiring a randomly picked answer forces your hand, so to speak. The selection of the answer is beyond your control, which is what “at random” means. Two coin flips (say) determine the outcome.

There does not appear to be ambiguity in “at random”, incidentally. There is no information beyond the information that there are four choices and only one must be picked. This (via Williams’s statistical syllogism) implies a “uniform” or maximum-entropy distribution for picking.

Anyway, to avoid the self-negation, the BSQE could be interpreted to mean, “If you were to suppose another student chose an answer to this question at random (but you must think it through and select one of A through D), what is the chance this other student is correct?”

We now have a problem: how many right answers are there? Is the BSQE structured like the Hamlet question? There is a similarity. The BSQE also has two identical answers, thus on the assumption that there is at least one right answer, A and D are either both right or they are either both wrong. B and C are different, so again they both cannot be right, though both can be wrong.

It is an assumption, though, that because the labels A and D are identical that if one is right both are. That assumption might not hold. Instead, the writer of the BSQE could have said “D is right and none other.” If this is so, then the chance the other student guesses correctly, given our evidence and assumptions, is 25%. And this is also so if the BSQE writer has said, “A is right and none other” or B or D. So if the BSQE writer meant only one answer was right and none other, then the chance the other student guessed correctly is 25%.

But you are not the other student. You must pick only one of A through D. We know the chance the other student guesses correctly is 25%, so you feel free to choose A or D. Choose A. If A was what the BSQE writer picked as the sole right answer, then you are fine. But if the sole right answer was chosen as D, then you are wrong, even though the label on A is the same as D. It would be as if you answered A for the Hamlet question but the teacher marked you wrong because he decided—and it was his choice—that only D was correct. Obviously, the same goes if you pick A but the BSQE writer picked D.

Then we are forgetting that the BSQE writer could have picked C or D. If so, you must choose C or D to have the right answer, even though the labels on these questions have no seeming bearing or relevancy. Again, it would be as if the Hamlet writer said obstinately decided that only C was right.

Thus, if the assumption there is only one right answer is correct, the BSQE implies A or D is correct (for you), but it might be that neither is. Self-referentiality has struck! No clear solution exists.

So change the assumption back and suppose that if A is correct then so is D. They have the same labels, after all. If the correct answer is B, then the probability the other student guesses correctly is 25%, and the same if the correct answer is C. If the correct answer is A or D, then the correct answer is guessed by the other student 50% of the time. We again must assume that the BSQE writer chose a right answer, which could be B or C or it could be either of A or D.

Equipped with this deduction (and it is a deduction), you answer B. If the BSQE writer chose B, then you are sitting pretty. But if he chose A/D or C, then you’re wrong. The same argument as last time applies for the other choices, and thus we are back deep into self-referentiality, the true vicious circle.

It’s here that we wade into deep kimchi. We skated around the question of how the BSQE writer chose his right answer last time, because we had no knowledge which unique letter he picked or how he picked it. To us, his choice was uniform, i.e. random. But supposing A/D are the same we have more information because of the duplication of labels. Did the BSQE writer constrain himself to all four or only three choices? Suppose just three, and he tacked on answer D as an afterthought (regardless whether A is correct). Or did the BSQE writer refer to your thinking process, assuming you would correctly deduce everything properly, and then pick his answer constrained by this assumption? The other student must still guess randomly in any of these scenarios, incidentally, so nothing changes for his guessing chances.

With the repeated reminder that the probability you deduce depends on the assumptions that you make, and thus that different sets of assumptions can lead to different probabilities, I leave the rest as reader exercises.

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1Thanks to Zach Woodmansee, HC, commieBob, and an anonymous person. Thanks to John Cook for pointing out a stupid typo.

25 Comments

  1. My random answer is 39.5%. Since that’s not on the list maybe true True answer is really 0%? If A, B & D are all correct then the answer would be 75% — also not on the list. I wonder. Does the writer of this question hand out probabilistic grades?

    Reminds me of those SAT questions where one is asked to pick out the different image from among four dissimilar images.

    Q: If I buy something from Random House isn’t that selecting something a Random? Will it be any good? Presumably, the House of Random offers only the best quality random. Must I answer the BSQE while at Random? Inquiring minds want to know.

  2. Meister Briggs,

    For the “Hamlet” question did you mean “A or B” as written, or “A or D? Also the link to the “example” which I very much wanted to read before thinking further, doesn’t work. I want to get all the evidence in, so to speak, before tackling this.

    Thanks!

  3. “If you choose an answer to this question at random,”
    Roll a die to choose, fives and sixes don’t count.
    “what is the chance you will be correct?”
    You have met the condition of random choice, you are correct, it does not ask if the answer you choose is correct.

    Strongly supports my theory that I am always correct anyhow.

  4. I recall a parlor trick, where someone is presented with a small strip of paper on which numerals are written as follows:

    1 2 3 4

    The person is asked to circle one number.

    He is then asked to turn the paper over, where he finds the written question:

    “Why did you pick 3?”

    While not successful every time, the frequency with which 3 is chosen apparently is far greater than would be expected if the process were truly random. I’ve speculated that there may be a tendency to avoid the extremes of 1 and 4, and perhaps it’s more natural for a right-handed person to circle 3 rather than 2.

    I realize this doesn’t directly relate to the blackboard question, except to the extent that making a “random” selection doesn’t imply that all four outcomes are equally likely if an individual is asked to make the selection at random. It would be ironic though, if it turns out that there is some subjective tendency to pick “C” 60% of the time.

  5. Assume one of the three percentages (25%, 50%, and 60%) is the right answer. Given that you choose at random (close your eyes and draw a ball out of a bag containing 4 balls),

    *if the right answer is 25%, then the chance of being correct is 0.5.
    *if the right answer is 50%, then the chance of being correct is 0.25.
    *if the right answer is 60%, then the chance of being correct is 0.25.

    Assuming that the three percentages are equally likely to be right, i.e., a 1/3 probability of being right, now apply the rule of total probability.

    So, if choose at random + assumptions above, P(being correct) = 1/3.

  6. The unexamined premise behind this question is that one is obliged to answer all asked questions. Even questions that cannot be answered. This is simply not so. If there is no honor in the question, there can be no obligation to answer nor even an to attempt an answer.

    The question is a clear attempt to place the target into a logical double bind game that he cannot win no matter how he answers. As such, it is a question without honor. The only way to win those kinds of games is not play them.

    There are several crisp comments that are appropriate for double bind questions. They usually contain explicit references to where the person asking the question should go and what they should do with their question.

  7. All generalisations are dangerous.

    My answser to the BSQE is 0%.

    To make this problem better, I changed anser c) to 0% when I asked this question of a friend.

    Annother way to weasle out: Why does random mean uniform? I could select a distribution such that C is correct.

  8. If you choose an answer at random, as directed, then all choices are correct. You have fulfilled the requirements of the question. Thus the correct answer is 100%, but this is not one of the answers given, let alone all of the answers. Thus the correct answer is 0%, but this is not one of the answers either. And so it goes.

  9. What is wrong with you people?

    BobN is right. It’s a cute little recursive ‘liars paradox’ problem. There is no answer.

    I cant tell if y’all are over- or under-thinking it.

  10. It is certainly not recursive. It is an undefined problem, in the sense that what is “correct” is never defined, and so can not have any probability of being right assigned.

  11. I hold in my hand a government publication (okay, I have set it down) that says on p. 96… “THIS PAGE LEFT INTENTIONALLY BLANK” (CAPS in the original). The page is manifestly not blank, but hey, it’s the government. The government says things like “the economy grew 2.5 percent in Q2” but do we believe that? Do they express it as a probability?

    Not in the “news” reports.

    Now, on to the BSQE!

    In Robert Heinlein’s marvelous story “By His Bootstraps” the protagonist is seemingly trapped in a self-referential (or recursive, you choose) Time Loop.

    “Time after time he had fallen into the Cartesian fallacy, mistaking clear reasoning for correct reasoning.”

    “If God created the world, who created God? Who wrote the notebook? Who started the chain? He felt the intellectual desperation of any honest philosopher. He knew that he had
    about as much chance of understanding such problems as a collie has of understanding how dog food gets into cans.”

    And just as Bob suddenly realizes who he is, I realized that I could with one Act of Will step out of the frame and be free, free of the Madness that almost got me as I contemplated the Question. I broke the chain.

    That’s what I got for trying to think of an answer before reading further.

    Thank you!

  12. At the risk of reading too much into the question, which contains two parts:

    The chance of being correct is “B”, 50%; which is the probability of selecting the value 25% which is the probability involved in randomly choosing one of four options?

  13. The ‘question’ does not appear in the picture (it may be a secondary question). If, however, it is self-referential, then the answer is 33% because there are only three choices and it does not matter how many times they are duplicated. Also, the author does not state whether you must choose ‘A’ through ‘D’ as your answer. It is a blackboard question- so, either raise your hand and say “33%” or just write down ‘33%’ for your extra credit answer.

  14. Everyone is assuming all the figures are in base ten. I make no such assumption. I assume all the answers are unique. Therefore one of the 25% is a different number base. You have only a 1 chance in 4 of randomly picking the answer containing the base 10 number 25%.

  15. I read the question to mean find “Ans” such that P(choosing Ans at random) = Ans

    This has no solution as
    P(25%) = 50%
    P(50%) = 25%
    P(60%) = 25%

  16. Many people seem to be arguing, with good reason, that the question is paradoxical or self-contradictory. but this is still utilizing the same assumption made by the OP: “assume that the right answer appears at least once in the BSQE.”

    In logical analysis, if you make an assumption, and this leads to a contradiction using the stated axiom(s) (in this case the BSQE itself), then the converse of the assumption must also be analyzed. If this too leads to a contradiction, only then can we say the original axiom(s) are logically false.

    So lets look at the converse: Assume that the right answer does not appear in the BSQE. Is there an answer that does not lead to contradiction? If the answer does not appear, then the chances of it being selected at random from those available is 0 in 4 or 0%. Because “0%” is not one of the given answers, there is a 0% chance you will be correct as well. This is logically consistent. A logically acceptable answer is 0%.

    On a side note, if “0%” were substituted for 60% for answer C, like Doug M suggested, then there would be a 25% chance to randomly select it and then we would have a logical contradition for both the assumption and its converse. Then the axiom (the BSQE) would be illogical.

  17. My answer is the following. I do not pick one of the multiple choice answers listed, as they are part of the question, which in the overall is not truly a multiple choice question.

    Thus I would write out.

    The chance of being right is 50% if I chose to answer the question randomly.

    (as there are 4 choices to chose from giving each choice a 25% chance of being right, and since there are two choices which state 25% is the correct answer that means half or 50% of the answers are correct, giving a 50% chance that a randomly selected answer would give the correct probability of being right.)

  18. I saw this at a tennis forum a few weeks ago and this is how I replied to it, for whatever is worth.

    Two main ways to look at it. First: The question refers only to itself in a loop. So this dialogue begins and never ends:

    Chance that I would be correct in what?

    In choosing an answer to this question at random.

    What question?

    This question: “What is the chance that you would be correct”?

    Correct in what”?

    In choosing an answer to this question at random.

    What question?

    This one: “What is the chance that you would be correct”?

    And so on, like in Who’s on First.

    But it has additional twist. If you assume that (regardless of whether the question makes sense or not), there should be ONE correct answer: A, B, C or D, then it would be 25%, but this is immediately eliminated because there are 2 replies with 25%, which would make your chance 50%.

    There is actually one reply that covers 50% , but since there is only one, that would make your chance 25% again. But there are two of those, so you are back to 50%, etc

    So the answer looks like 0%. But that’s not covered. And if it were covered, it would no longer be 0 percent anyway.

  19. I think your all nuts. It is a statement referring to a question that we do not know. It is simply stating that in multiple choice questions not knowing the answer and guessing will be right by the percentage shown. No it does not have to add up to 100%. I was told years ago when you don’t know the answer choose C as that has the greatest chance of being correct.

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