[roolstar]

Well this always bugged me!

A math teacher declared one Friday: "I will give you a drop quiz next week, and none of you will know which day it will be so prepare for every day!"

Now one of the brilliant students started reasoning on Saturday:

- We cannot have the drop quiz next Friday, otherwise we will all know for sure next Thursday (since we did not have the drop quiz yet) that the drop quiz will be on Friday and this will contradict the teacher statement!

- We cannot have the drop quiz next Thursday, otherwise we will know on Wednesday (since we did not have the drop quiz yet) that the teacher is left with no choice than to give us the drop quiz on Thursday, since he cannot give it on Friday (above)

- We cannot have it on Wednesday then for the same reason like above

- Nor on Tuesday

- Nor on Monday for that matter!!!

So in short, the teacher CANNOT maintain his promise!!

Still, the teacher gave them a drop quiz on Tuesday and nobody was prepared for it nor knew about it the day before!!

What's the error in this reasonning????

PS: I don't really have an answer but I'm really curious about your reasonning and maybe I can stop stressing about this!!

INTERSTING FACT: This situation actually took place during World War II when a radio station (british I think) announced that they will have a very high profile guest on one of their show during a certain month (April I think). But due to the sensitivity of the situation, they cannot annouce the actual date of the interview for security reasons. They cannot have people knowing the date before the interview. So a Mathematician send them a telegram saying: "This cannot be done STOP"

[Guest]

It's false dilemma using a kind of slippery slope. It's a pretty common logical fallicy. You assume that because it's true that a Friday quiz would be given away by that knowledge on Thursday, the same must be true of a Thursday Quiz given away Wed, and so on. In fact, the only reason that Friday is given away is by process of elimination. That giveaway cannot happen without the actual elimination.

On Sunday, there is a set of 5 possibilities.

On Monday, 4

Tuesday, 3

Wednesday, 2

Thursday, 1

Therefore as the week progresses you become more certain of the outcome and can guess with higher and higher probability, but you cannot backward engineer that process of elimination to say that because X is true on Thursday, it must also be true on Wednesday.

[Guest]

This is probably wrong as hell, but here's my two cents:

If the test had been given on Friday, the students would not have known it was on Friday until after class Thursday. Assuming no test had been given Monday, Tuesday, or Wednesday, there was a 50/50 chance (based on the students knowledge, not the professors) that the test could have been on Thursday or Friday. Keep backing up and there's a 33(.33333333333333333333333333333333333333333333333333333)% chance that it could have been Wednesday, 25% for Tuesday, and 20% for Monday.

The base assumption is that if the test is not on Monday, Tuesday, Wednesday, or Thursday then it will be Friday and so can't be Friday because they would know about it. But since they do not know the day of the quiz they also cannot know what day the quiz isn't. Therefore, they cannot make the first assumption that early in the week.

[roolstar]

It's false dilemma using a kind of slippery slope. It's a pretty common logical fallicy. You assume that because it's true that a Friday quiz would be given away by that knowledge on Thursday, the same must be true of a Thursday Quiz given away Wed, and so on. In fact, the only reason that Friday is given away is by process of elimination. That giveaway cannot happen without the actual elimination.

On Sunday, there is a set of 5 possibilities.

On Monday, 4

Tuesday, 3

Wednesday, 2

Thursday, 1

Therefore as the week progresses you become more certain of the outcome and can guess with higher and higher probability, but you cannot backward engineer that process of elimination to say that because X is true on Thursday, it must also be true on Wednesday.

You are right

But still, If we get to Wednesday without a drop quiz, the students can safely say: "The teacher HAS to do the test tomorrow (Thursday), otherwise we will KNOW that it's going to take place on Friday and his statement was incorrect!

And if we get to Tuesday, We know that his only choice is on Wednesday to aviod the above issue...

I know this reasonning is incorrect, but I'm looking for a simple approach to explain the logical flaw in it!

The fact that Thursday is 1 according to your method means that Thursday can be considered like Friday and be eliminated, this will eliminate Wednesday sort of speak... Or does it?

Edited February 9, 2008 by roolstar

[roolstar]

This is probably wrong as hell, but here's my two cents:

If the test had been given on Friday, the students would not have known it was on Friday until after class Thursday. Assuming no test had been given Monday, Tuesday, or Wednesday, there was a 50/50 chance (based on the students knowledge, not the professors) that the test could have been on Thursday or Friday. Keep backing up and there's a 33(.33333333333333333333333333333333333333333333333333333)% chance that it could have been Wednesday, 25% for Tuesday, and 20% for Monday.

The base assumption is that if the test is not on Monday, Tuesday, Wednesday, or Thursday then it will be Friday and so can't be Friday because they would know about it. But since they do not know the day of the quiz they also cannot know what day the quiz isn't. Therefore, they cannot make the first assumption that early in the week.

Is it really a 50/50 chance for Thursday and Friday?

What makes me ask that is the fact that the teacher simply cannot give the drp quiz on Friday otherwise everybody WILL know about it the day before. And this means his statement that NOBODY will know before hand would be incorrect!

So Probaility of Friday = 0% and now Thursday has a 100% chance ==> The teacher cannot wait till Thursday either ==> probility of Thursday becomes 0 and P Wednesday becomes = 100% and so on...

The term Probability used here isn't really Probability in the scientific sense as in "favorable possibilities / total possibilities"; it means Possibility of making the test on a certain day. P(Friday)=0%

Edited February 9, 2008 by roolstar

[Guest]

Is it really a 50/50 chance for Thursday and Friday?

What makes me ask that is the fact that the teacher simply cannot give the drp quiz on Friday otherwise everybody WILL know about it the day before. And this means his statement that NOBODY will know before hand would be incorrect!

So Probaility of Friday = 0% and now Thursday has a 100% chance ==> The teacher cannot wait till Thursday either ==> probility of Thursday becomes 0 and P Wednesday becomes = 100% and so on...

The term Probability used here isn't really Probability in the scientific sense as in "favorable possibilities / total possibilities"; it means Possibility of making the test on a certain day. P(Friday)=0%

Here's the flaw in that logic: Assuming the professor doesn't want them to know the quiz is on Friday, the probability of the quiz being Thursday doesn't go to 100% until AFTER Wednesday's class is over. Before that, it's only 1 divided by the number of days until Thursday percent chance (i.e. 25% before Monday, 33% before Tuesday, and 50% before Wednesday's class)

[Guest]

I'll just post the flow of my thoughts and see if I can come up with anything. (afterwords edit...: Yay, i think i got it. Kinda.)

Monday Probibility - 14% chance after the previous day

Tuesday Prob. - 28% chance.(posibly)

Wednesday Prob. - 42% chance.(posibly)

Thrusday Prob. - 85% chance.(too high! predicable)

Friday Prob. - 100% chance.(too high! predicable)

Monday notes: Unlikely. Too early in the week - does not allow time to study.

Tuesday notes: Unsure. Not much to say. Seems like an okay day to have it.

Wednesday: Unprobable. It's in the exact middle of the week, so any student might just look at that and think this will be the most likely day to have the test.

Thursday: Unlikely. It get's more likely, because the chances of the the teacher putting it on friday are slim. But some students might anticipate that. Gives students too much time to study for the test.

Friday: Definitly unlikely. Way to predicable on Thursday. Also gives students too much time to study.

Seeing this, Thursday, Friday, and Monday are probobly out. That leaves Tuesday and Wednesday. Okay, so that should do it...

Tuesday and Wednesday are the best guesses. The teacher made a good decision for picking Tuesday, because if the test was on Wednesday, it would be to predicable, since a)its in the middle of the week b)after Tuesday then the test would have to be on Wednesday. This forces students to study on Monday. It's the perfect balance between predicability and days to study.

Still, none of my or anyone elses reasoning could predict this exactly. The teacher can pick any day - even Friday, just to make the students study more. Actually, Monday is the least likely choice, because it would be a very bad teacher to do that to the poor students. Truly, in a perfect world I think it would be Tuesday (which aparent it was...), but really, against all the predictions, Monday would be the most unpredictable. But it would be mean to do that.

[Guest]

I thought this was as obvious riddle, but then when i started writing my answer i proved myself wrong as i started to write the reasons it couldnt be on the days. If you work forwards though from monday to friday then it can be on any possible day except friday.

[Guest]

I thought this was as obvious riddle, but then when i started writing my answer i proved myself wrong as i started to write the reasons it couldnt be on the days. If you work forwards though from monday to friday then it can be on any possible day except friday.

Personally, I think, rather than friday, that the drop quiz could be on any day except Monday. This would really ruin the point of having such a random quiz day, and it would be very unfair. Besides, the math teacher wants to get the students to study every day, and anticipate that the test is next day.

Every day after Monday that the quiz is not on will make the chances higher of the quiz being the next day, and the next, etc. That would basicly mean that there isn't a real way to predict it - instead, the best prediction for a student is to study every day. Although Tuesday is a good choice, because forcing students to study for a quiz for a week (if the quiz was on Friday), when it's not even 4 days until it would be cruel. And it ruins student predictions.

[bonanova]

I first read this paradox in Martin Gardner's "The unexpected hanging ..."

A prisoner was sentenced by a judge to hang, the following week, on a day he could not predict.

His lawyer reasoned like the brilliant student here, and he convinced the prisoner he was safe from execution.

On Wednesday, however, the hangman executed him.

Much to his surprise.

To analyze, consider the judge saying simply, "you will be executed next week on an unannounced day."

Now the prisoner fears for his life each day.

Adding the constraint "you can't figure out which day" really adds nothing to the situation.

It means only that the Judge makes the decision independently.

We can make it simpler.

The judge says "You will be executed within one day. You can't know what day that will be."

The lawyer says, well, if it's within one day it must be tomorrow. Since we know it must be tomorrow, it can't be tomorrow.

The prisoner is executed tomorrow - unexpectedly.

The reasoning of safety is unreliable: certainty that it won't happen makes the happening unexpected and therefore permissible.

The operative statement is "you will be executed within one day."

The rest is a red herring.

[roolstar]

I first read this paradox in Martin Gardner's "The improbable hanging ..."

A prisoner was sentenced by a judge to hang, the following week, on a day he could not predict.

His lawyer reasoned like the brilliant student here, and he convinced the prisoner he was safe from execution.

On Wednesday, however, the hangman executed him.

Much to his surprise.

To analyze, consider the judge saying simply, "you will be executed next week on an unannounced day."

Now the prisoner fears for his life each day.

Adding the constraint "you can't figure out which day" really adds nothing to the situation.

It means only that the Judge makes the decision independently.

We can make it simpler.

The judge says "You will be executed within one day. You can't know what day that will be."

The lawyer says, well, if it's within one day it must be tomorrow. Since we know it must be tomorrow, it can't be tomorrow.

The prisoner is executed tomorrow - unexpectedly.

The reasoning of safety is unreliable: certainty that it won't happen makes the happening unexpected and therefore permissible.

The operative statement is "you will be executed within one day."

The rest is a red herring.

Not even if the teacher said: "You will have a drop quiz next week in a way that you will never know for sure it's the next day!"?