In Defense of Ambiguous Problems
Yesterday, I posted the Curious Prisoner Puzzle. The puzzle was structured to attempt to fool people into answering 1⁄2, which is possible, but certainly not unambiguously true. Any answer has to acknowledge the ambiguity, but at least in my opinion, the best way to reinterpret the problem to ensure that there is a single, unambiguous answer is as follows: Assume that you are told the statement when it is true and nothing is otherwise. Then, the answer is 1⁄3. Well done to all the commentators who quickly managed to figure all this out.
So why did I post an ambiguous problem?:
If the point of these puzzles is to teach a lesson, dodging an incorrect answer suffices. I actually couldn’t see a way of making the problem unambiguous without giving the game away with regarding the answer not being 1⁄2
You can’t assume that every problem you’ll try to solve will turn out unambiguous (particularly if you study philosophy), so it’s useful to be able to recognise when this is the case. If you always practise with unambiguous problems, then you are biasing yourself to assume problems are unambiguous, which is why people often struggle with this
This problem demonstrates the importance of counterfactuals in terms of probability. In fact, an ambiguous problem is probably the best way to demonstrate that such problems are ambiguous without defining the counterfactuals. In particular, it shines a light on problems like the classic, “What is the chance that two kids were both born on the same day of the week if at least one of them was born on a Tuesday?”
Further ambiguous problems can reveal the implicit assumptions made elsewhere. Suppose we have are four boxes: red, green, blue and white. One is picked at random to contain a prize, then the host peeks and tells you that it isn’t in the white box. What is the chance that it is in the blue box? This problem is mathematically the same, yet we generally don’t say that it is ambiguous even though the counterfactual hasn’t been specified. That’s probably because the different natural interpretations of the host’s algorithm all give the same answer of 1⁄3, but this is an important realisation because it affects a massive proportion of probability problems:
The host randomly picks a color that doesn’t contain the prize and tells you that it isn’t in that box
The host randomly pick a color and tells you if that box contains or doesn’t contain the prize.
The host always tells you if the white box contains or doesn’t contain the prize.
If the white box doesn’t contain the prize host tells you that the white box doesn’t contain it. They further tell you that they were always going to tell you if this was the case and that they weren’t going to say anything otherwise.
As per the previous, but with a random box