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OT: A variant that might be a little more interesting

Posted By: Timothy Chow
Date: Wednesday, 2 November 2011, at 5:38 p.m.

In Response To: OT: Greatest Probability Problem? (Steve Mellen)

I agree that this isn't a very interesting problem as stated. However, Stein Kulseth's variant got me thinking. How about if there were six options, as follows?

A. 1/2
B. 1/2
C. 1/2
D. 1/3
E. 1/3
F. 1/6

Here it seems at first that any of 1/2, 1/3, and 1/6 leads a self-consistent result. However, suppose we now invoke the following multiple-choice principle (MCP) that seems to be tacitly accepted by most people:

MCP: In a multiple-choice question, one and only one answer is correct.

It then follows that F must be the correct answer. (Of course, given MCP, F must be the correct answer regardless of the question...)

Whether or not the above variant is interesting, I think it sheds some light on the original puzzle. The trouble is that there's no way to answer the original puzzle in a way that is consistent with MCP, without resorting to quibbling or trickery of some sort. Now, if we don't insist on MCP, then 0% becomes the unique self-consistent answer to the original puzzle. But that's not very satisfying, so the original puzzle still counts as uninteresting in my book.

It's still conceivable that an interesting puzzle can be constructed using this circle of ideas, but it would require some additional creativity.

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