First, you unexpectedly switched from the unexpected hanging to the unexpected test in your third last paragraph :)
Second, surprise is best defined as inaccurate map, and the judge/teacher in their pronouncement assumes that the prisoner/student will not be able to come up with an accurate map. If the prisoner can come up with one, then the judge’s assertion of “it will be a surprise” will be just another inaccurate map, not the territory. The two maps cannot be both accurate given the stipulation of “surprise”.
The prisoner’s reasoning, as described, is a maximally inaccurate map.
What would be a maximally accurate map for the prisoner? That crucially depends on the mechanism the judge uses to decide on the day. If the judge rolls a five-sided fair die, then the odds are 20% on Monday, 25% by Tuesday, 33% by Wednesday, 50% by Thursday, 100% by Friday. If the judge instead flips a coin before each day, the probability is 50% each day except on Friday, when there is no coin flip and it’s 100%. If the judge instead decides that Friday is right out and rolls a 4-sided die, then it’s 25%/33%/50%/100%/0%. Maybe the judge always schedules executions on Wednesdays, and if the prisoner knows that, then the odds are 0/0/100%.
Can the prisoner construct an accurate map? Who knows, their capabilities and their knowledge of the judge are not specified in the problem statement. Either way, increased accuracy of one map can only come at the expense of the accuracy of the other map. That’s all there is to it.
First, you unexpectedly switched from the unexpected hanging to the unexpected test in your third last paragraph :)
Second, surprise is best defined as inaccurate map, and the judge/teacher in their pronouncement assumes that the prisoner/student will not be able to come up with an accurate map. If the prisoner can come up with one, then the judge’s assertion of “it will be a surprise” will be just another inaccurate map, not the territory. The two maps cannot be both accurate given the stipulation of “surprise”.
The prisoner’s reasoning, as described, is a maximally inaccurate map.
What would be a maximally accurate map for the prisoner? That crucially depends on the mechanism the judge uses to decide on the day. If the judge rolls a five-sided fair die, then the odds are 20% on Monday, 25% by Tuesday, 33% by Wednesday, 50% by Thursday, 100% by Friday. If the judge instead flips a coin before each day, the probability is 50% each day except on Friday, when there is no coin flip and it’s 100%. If the judge instead decides that Friday is right out and rolls a 4-sided die, then it’s 25%/33%/50%/100%/0%. Maybe the judge always schedules executions on Wednesdays, and if the prisoner knows that, then the odds are 0/0/100%.
Can the prisoner construct an accurate map? Who knows, their capabilities and their knowledge of the judge are not specified in the problem statement. Either way, increased accuracy of one map can only come at the expense of the accuracy of the other map. That’s all there is to it.
“First, you unexpectedly switched from the unexpected hanging to the unexpected test in your third last paragraph :)”—fixed now