There is an enormous (far too enormous for its value to the world, in my opinion) literature on the unexpected hanging paradox (also known as the surprise exam paradox) in the philosophy and mathematics literature. The best treatments are:
far too enormous for its value to the world, in my opinion
The paradox actually has practical implications. It shows a general mechanism by which you can “surprise” someone despite a predictable outcome. It goes like this:
1) You tell someone they will be “surprised” by an upcoming event (e.g., what gift you will buy them). 2) They start to suspect it will be one of a number of unusual outcomes. 3) The event actually has its regular, boring, predictable outcome. 4) But the other person is still surprised, since they did not expect the boring outcome (when before your statement, they did)!
I know of a major case where this reasoning was applied: on one season of the TV show The Apprentice. (The show where people try to get a job with Donald Trump and one person is eliminated from consideration [“fired”] each week.) During the second episode/competition, one contestant walked out due to frustration, and she didn’t come back until evaluation time.
Then, in ads for the next episode, they said, “Next time, on The Apprentice, one candidate will quit the competition—and you’ll never guess who it is!”
This, of course, prompted speculation that someone other than the last episode’s quitter would be the one to quit … but no, it was the same woman who left, this time permanently, rather than being fired. Well, it was certainly a suprise by that point!
When I tried to solve this problem about 10 years ago, I came up with the equivalent of Fitch’s “Goedelized” solution, described on pages 5-6 of Chow’s article. I’m still not sure why many people consider it wrong; it seems to utterly dissolve the “paradox” for me.
There is an enormous (far too enormous for its value to the world, in my opinion) literature on the unexpected hanging paradox (also known as the surprise exam paradox) in the philosophy and mathematics literature. The best treatments are:
Timothy Y. Chow, The surprise examination or unexpected hanging paradox, American Mathematical Monthly 105 (1998) pp. 41-51. (ungated)
Elliot Sober, To give a surprise exam, use game theory, Synthese 115 (1998) pp. 355-373. (ungated)
The paradox actually has practical implications. It shows a general mechanism by which you can “surprise” someone despite a predictable outcome. It goes like this:
1) You tell someone they will be “surprised” by an upcoming event (e.g., what gift you will buy them).
2) They start to suspect it will be one of a number of unusual outcomes.
3) The event actually has its regular, boring, predictable outcome.
4) But the other person is still surprised, since they did not expect the boring outcome (when before your statement, they did)!
I know of a major case where this reasoning was applied: on one season of the TV show The Apprentice. (The show where people try to get a job with Donald Trump and one person is eliminated from consideration [“fired”] each week.) During the second episode/competition, one contestant walked out due to frustration, and she didn’t come back until evaluation time.
Then, in ads for the next episode, they said, “Next time, on The Apprentice, one candidate will quit the competition—and you’ll never guess who it is!”
This, of course, prompted speculation that someone other than the last episode’s quitter would be the one to quit … but no, it was the same woman who left, this time permanently, rather than being fired. Well, it was certainly a suprise by that point!
Yeah, came here to say the same. Thanks.
When I tried to solve this problem about 10 years ago, I came up with the equivalent of Fitch’s “Goedelized” solution, described on pages 5-6 of Chow’s article. I’m still not sure why many people consider it wrong; it seems to utterly dissolve the “paradox” for me.