“Probably” Correct

Nerd Sniping

Today I saw a brilliant little bit of nerd-sniping on Facebook;

If you were to choose the answer to this question randomly, what is the probability that you would get it right?

A) 25%

B) 50%

C) 60%

D) 25%

Oh boy! So let’s delve into why this is a perfect piece of nerd sniping, shall we? There are “two” potential resolutions to this question;

1) 25%

If there is exactly one unique answer to the question, then when you pick an answer at random from four answers, then the chance of picking the answer designated as “correct” is 1 in 4, or 25%. Easy! Except if you know that the correct answer is in two locations, then;

2) 50%

If you know that two of the answers are exactly the same, then you know that (if this is allowed) it is correct and the chance is 50%. In other words; by accepting answer 1) and looking at the answers, you realise that 25% occurs in two locations and so the chance of selecting 25% is 2 in 4, or 50%! OR…

3) 50%

If you disallow potential answers that aren’t unique then you have to account for the fact that neither of the 25% answers actually count as an answer; the choice is then between the two *valid* answers (assuming one of them is then correct). This is a chance of 1 in 2 or 50%!

HOWEVER if you now know that the answer is 50%, then 25% is no longer the correct answer to the question. Now that you know this, and that the correct answer occurs once in four (i.e. 25%).

Even Jim Al-Khalili is puzzled!

OMG Paradox!

Not so fast! Like with all paradoxes, there’s something wrong here. Let’s find out why;

Firstly, option 3) that I presented actually prevents the paradox from arising. Since you discount answers which are the same, you have a logical choice between two and hence there is no paradox; the answer is 50% and the chance of picking it is also 50%. Correct! Alternatively, in the case 3) if neither of the answers is correct then the chance of picking randomly the correct answer is zero, in which case it’s a trick question and you get off without answering it.

However, we’re still left with a situation where knowing the answer is 25% changes the answer to 50% and vice versa. How do we resolve this? Let’s consider a little mathematical trick. The expectation value of a test like this can be defined as the sum of each result multiplied by it’s probability. We have two potential results;

a) The correct answer is 25% with a probability of 50%

b) The correct answer is 50% with a probability of 25%

Now calculate the expectation value; (0.5×25)+(0.25×50) = 25! In other words, when you treat this as an experiment where you can get either result a or result b you find that the average result is 25%! That is; the answer to the question, on average, is 25%.

So 25% is correct?

Probably not!

Technically the probabilities in the sum I used above have to add up to one, and of course 25% + 50% = 75%. 25% isn’t the correct answer, but then that’s not what an expectation value is anyway. So what I’m saying is that 25% could well be the correct answer if there even is one, but we have no idea since the question is poorly defined. That’s more or less the definition of a paradox in the first place.

About stoove

A physicist, researcher, and gamesman. Likes to think about the mathematics and mechanics behind all sorts of different things, and writing up the thoughts for you to read. A competent programmer, enjoys public speaking and mechanical keyboards. Has opinions which might even change from time to time.
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