Mensa Math Paradox

Mensa is a high IQ society. The Facebook groups are quite odd to participate in. They mostly demonstrate that you can have a high IQ and be a good person, or a bad person, an informed person, or an uninformed person. Often, a high IQ just gives you the ability to confuse yourself more. High fluid IQ means that you have the capacity to recognize patterns faster than the average person. High concrete IQ means that you know more of these patterns in a given society than the average person. But, I digress. Here is a math problem that seemed to stump everyone in the group, except for me (I think that's just the right amount of hubris).

Someone posted this problem and called it a math paradox.

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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%

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Go ahead and give your answer and then see if you think I'm right after you read the rest of this post. I take no offense if you disagree with me. Here was my initial response to the problem.

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Jeff - I'm not a mathematician, but this seems pretty simple. If we have 3 answers (because of the repeat) and we randomly choose an option with the inbuilt assumption that one is correct then we have: a 100 percent chance of being right or wrong, a 66 percent chance of being wrong, and a 33 percent chance of being right.

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I just left it at that. I figured people would kind of put the rest together on their own. Rod responded to my comment and said this.

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Rod - but at random, we'd choose (A)25% or (D)25% 50% of the time. If that's the right answer, then the odds are

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A lot of people said something similar, but I think they are making a mistake.

I think people are reading this:

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

like this:

If you choose an answer to a question that has four possible answers at random, what is the chance that you'll be correct?

In that case we all know it is 25 percent. So we look at the answers knowing that there is something tricky about it or it wouldn't be a post... and we find it. There are two correct answers! That means we have a 50 percent chance. But that's not the actual question, it's what we are thinking the question is.

If we are going to substitute a phrase we should read it like this:

If you choose an answer to a question that has three possible answers at random, what is the chance that you'll be correct?

Because that's all of the possible answers there truly are. Now we know that they have not given a possible answer that satisfies the question. Thus, what we have is a false dichotomy (meaning neither answer is correct), but we could call it a false quadchotomy (meaning none of the four answers is correct).

The correct answer is either 1 - E) None of the above, or 2 - To dismiss the question as a logical fallacy, which I have called a quadchotomy. (As a side note, using a dichotomy is a common sales technique, not to mention politics. Would you like this one or this one? Neither please, thank you. Would you like to do X or Y? Actually, I would like to do Z.)


I've written three fictional pieces that I like so far.

"The City of Peace" - A future history science fiction utopia/dystopia action adventure in a framed story of a father telling his son a story about the child's grandfather. That was a crazy sentence.

"The Birth of Hanniba'al" - A dark, somewhat alternative, historical origin story for the Carthage General Hannibal.

"Matt's Eyes" - Don't read this if you don't like horror stories.

Here are three of my most popular posts.

"The Making of a Great First Line in Fiction"

"A Letter to My Niece in 2034"

"The Most Important Question in Philosophy - Part 4 of 4"

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