Multiple Choice Probability Puzzle

   
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A Multiple-Choice Probability Puzzle - 2011/11/15

Recently the following puzzle was running around the 'net:

If you choose an answer
at random, what is your
probability of being
correct?

  • A: 25%
  • B: 50%
  • C: 60%
  • D: 25%

The immediate thought is - there are four options, so if I pick one at random then each has a one in four chance of being chosen. That means the answer is 1/4, or 25%.

But hang on, 25% turns up twice, so if the correct answer is 25% then there's a 50% chance of picking an option with that answer. So then the answer is 50%, not 25%. So 25% can't be the right answer.

Well, is 50% the right answer? There's only one of them, so the chance of picking that is one in four, which is 1/4, which is 25%. So if 50% is the right answer, there's a 25% chance of picking it. So 25% is the right answer, and that doesn't work.

And 60% is just a complete non-starter.

So what is the right answer?

Lots of people tweeted me about this, asking what the right answer is, and commenting that their brains were melting, exploding, or otherwise acting in an unpleasant or uncomfortable manner.

This "fifty/fifty" answer really makes maths teachers despair. After all, probability and statistics is all about the observation that not everything happens with equal likelihood.

Some people say that the chances of winning the lottery are 50:50 - either you do or you don't. But the technical interpretation of the expression "50:50" is that it happens, on average, half the time. clearly you won't win it every other week, so there's more going on, and probability and statistics helps us make sense of that.

One person tweeted that either you got the right answer or you didn't, so it was fifty/fifty. Clearly not very helpful, but having said that, in this case there is a trick answer relying on this idea. The original question didn't say that we pick an answer uniformly at random - we could choose answer B half the time, answer C the other half of the time, and answers A and D not at all. Then the right answer is 50%, and that works.

Similar "uneven" choices can be made to work in the other cases, and it's interesting (and difficult) to explore the possibilities. I'll leave it as an exercise for the interested reader to find distributions that make any given answer "right." There's even a way to make three of the options right all at the same time, although that really doesn't sound right at all!

It works, although some people feel that it's cheating, and somehow wrong, not to choose among the options with equal probabilities.

So here's another answer.

Since none of the options seem to work, then none of them can be right. Since you will choose one of them, you will never get the right answer.

Hence the probability of being right is 0. That option doesn't appear, so if you pick randomly you won't get it, and so your chance of being right is 0%, and so it's consistent.

So that's another possible answer.

  • A: 20%
  • B: 40%
  • C: 0%
  • D: 20%
  • E: None of the above.
Of course, the problem was supposed to be mind-bending, and we can now prevent this second answer by changing the options. We can instead have the options shown here at right.

Now the second suggested solution - the 0% solution - doesn't work, because if we choose uniformly at random there is a non-zero chance that we pick that. So we're back to the non-uniform choice idea, and the many possibilities afforded by it.

But underneath all this is another question:

Why should the question have a "right" answer at all?

So many of the tweets I got were indignant that there didn't seem to be a single, clear, correct answer. But life's not like that. Some questions don't have answers, and it's especially difficult when things are self-referential.

We can create impossible situations.

People call them paradoxes, but the term is abused and overused. In some cases it's better simply to say that it's ill-posed, undefined, or otherwise impossible.

Questions like that are great for exploring the limits of logic and the recesses of reason.

And they make my brain hurt.

That's their job.

Other examples of paradoxes (or questions that are ill-posed, or something) include:

  • The Paradox of the Unexpected Hanging
    • ... which explores what it means by "To Know"
  • The Banach-Tarski paradox
    • ... which in truth is just a theorem that shows that
      the real numbers are surprisingly unintuitive
  • Russell's paradox
    • ... which caused all of Set Theory to be re-worked

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Comments

@snapey1979 said: Nice blog. Just because a problem is easily stated, does not mean it is logically consistant, well posed, or soluble.
-- http://mobile.twitter.com/snapey1979

That pretty much sums it up, and is related to the "2=4" proof here:

I've decided no longer to include comments directly via the Disqus (or any other) system. Instead, I'd be more than delighted to get emails from people who wish to make comments or engage in discussion. Comments will then be integrated into the page as and when they are appropriate.

If the number of emails/comments gets too large to handle then I might return to a semi-automated system. We'll see.

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