惊讶测验悖论

惊讶测验悖论

原文

Albert announces to his class that he plans to surprise them with a quiz sometime next week.

His students first wonder if the quiz could be on Friday of next week. They reason that it can’t: if Albert didn’t give the quiz before Friday, then by midnight Thursday, they would know the quiz had to be on Friday, and so the quiz wouldn’t be a surprise any more.

Next the students wonder whether Albert could give the surprise quiz Thursday. They observe that if the quiz wasn’t given before Thursday, it would have to be given on the Thursday, since they already know it can’t be given on Friday. But having figured that out, it wouldn’t be a surprise if the quiz was on Thursday either. Similarly, the students reason that the quiz can’t be on Wednesday, Tuesday, or Monday. Namely, it’s impossible for Albert to give a surprise quiz next week. All the students now relax, having concluded that Albert must have been bluffing.

What, if anything, do you think is wrong with the students’ reasoning?

译文

艾伯特老师宣布，他下一周会给同学们来一次“突击”测验。

他的学生首先开始考虑能不能是周五，结论是否定的：如果测验放在周五，也就是前 4 天都没有测验的话，那么周四下课的时候同学们就知道测验一定是在周五的，这场测验就算不上“突击”的了（1）。

接下来同学们又开始向前推，他们发现也不可能是星期四：（1）处已经证明了不能是在周五测验，如果放在周四测验的话，那么在周三下课的时候同学们就会知道肯定是在周四测验了——周一周二周三没测，周五不可能，只能是周四了，这场测验就算不上“突击”了（2）。形似地，同学们接着往下推，不可能是周三 -> 不可能是周二 -> 不可能是周一。也就是说，艾伯特老师不可能给我们一次突击测验，他是在吓唬我们！

你认为学生的想法错在哪里？

问题所在

学生们的推理存在问题，关键在于他们忽略了反向推理的复杂性。具体来说，他们把“测验的日期不能被确定”与“测验不可能发生”混淆了。

• 在推理过程中，学生假设一旦排除了一天，他们就能确定测验的日期。这个想法是错误的。尽管他们每次排除了某一天，但实际上，Albert 给出“惊讶测验”的前提就是，学生在测验前无法预知日期。即使学生们排除了某些日期，Albert 依然可以在最后一天给出测验，这仍然可以是一个惊讶。

参考

https://www.cnblogs.com/liqiuhao/p/7785034.html