I was looking at this brainteaser and it doesn't make too much sense to me.
http://www.ibankingfaq.com/interviewing-brainteasers/three-envelopes-are-presented-in-front-of-you-by-an-interviewer-one-contains-a-job-offer-the-other-two-contain-rejection-letters/
Let's change the scenario. What if I initially had 1,000,000 envelopes to choose from. My chance it 1/1M. I pick one, they remove 999,998 leaving two on the table. I know the offer is in one of those two. If I stay with original one, what they are saying is that my chance remains 1/1M. Since I KNOW for fact that it's one of those two, wouldn't the probability automatically recalculate itself, and thus my new chance is 1/2 since now I only have two envelopes? That being said, it seems to me that no matter if I switch at the end or not, I still have 1/2 or 50% chance to pick the one with a job offer, since there are only two of them now.
Can anyone explain this to me?
Thank you for all comments.
Unless it's just not explained well, I think the logic in that brainteaser is bunk. You initially have a 1/3 chance of picking the offer letter. Once one of the rejection letters is revealed, your chances change to 1/2, so you are indifferent between the remaining 2 letters.
The brainteaser says that if you "switch", the prob. changes to 2/3, which to me makes no sense considering there is only 1 offer letter. The 2 rounds of "choices" are discrete probabilities.
this is called the monty hall problem. Wikipedia it for an expansive explanation. http://en.wikipedia.org/wiki/Monty_Hall_problem
Very interesting. The problem with the rejection letter question as it is phrased, is that it doesn't really convey the intent and knowledge of the interviewer. If the interviewer knows the contents of each letter, and therefor chooses to reveal a rejection letter, then the probabilities are no longer discrete. Under the assumption that the interviewer does not know the contents of each (and could therefor reveal the offer letter after the first choice, leading to an immediate rejection), then I think my above solution would apply.
yeah, it's a well known mathematical puzzle. You can also just list out all the possible outcomes: When there are 3 letters, you either pick rejection 1, rejection 2, or job offer. -If you picked rejection 1, the interviewer would take away rejection 2. The other choice is the job offer -If you picked rejection 2, the interviewer would take away rejection 1. The other choice is the job offer. -If you picked the job offer, the interviewer would take away either rejection 1 or 2. The other choice is a rejection. So 2/3 of the time, the other choice is the job offer, 1/3 of the time your original choice is the job offer.
Thanks for the link, I read it and it makes sense now.
If I was an interviewer, I'd never ask this question, because most people would probably get this wrong and then I'd have to waste 15 minutes out of the 30 allocated for the interview, just to explain the reasoning behind the correct answer :-)
I think ehf3660 is right. (and I am right too :-) If the question doesn't specifically state that the interviewer knows which envelope is which, then the probability changes to 50/50 and it doesn't matter if you switch or not.
TH73, you're wrong. It doesn't matter if the interviewer knows the contents of the envelope or not. The only way that would change the problem would be the introduction of the possibility that he would open the offer letter instead of one of the rejections. But if a rejection is opened, you would still want to switch because it would double your chances of getting the offer.
Think of it this way. Three envelopes, one job offer and two rejections, are split into a pile of one and two. Which pile is more likely to have the job offer. THE BIGGER FUCKING PILE, which is all this question essentially is.
Well said gregweinstein.
except what gregweinstein said is completely wrong. after the interviewer has opened one envelope, both piles have one letter. so neither pile is BIGGER. but you should switch as Simian and xxcobra02 explain above.
No. By switching you are effectively choosing the larger pile. Opening the envelope is irrelevant since the pile of two will always contain at least one rejection letter (i.e opening a letter adds no new information). All that matters is the pile of one has a 1/3 chance having the offer while the pile of 2 has a 2/3 chance (discrete random variable). So to further simplify, the question is basically asking if you rather pick two envelopes or one. Think about it.
Choosing the other pile is indeed the correct answer. Hopefully I can offer a different angle on it:
I think we can all agree that when the chooser first picks a letter, he has a 33% chance of choosing the right one. In a standard probability question, should another letter be pulled from the pile and found to be a rejection letter, the probability would increase to 50%. What makes this problem unique is that, of the remaining two letters, a rejection letter is ALWAYS pulled. By always pulling a rejection letter, we do not actually gain any further insight as to whether or our original pick was correct or not. Our chance of having picked the correct letter originally remains at 33%.
Now to calculate the probability that the remaining letter is not a rejection. This was done well above but I'll reiterate. When you make your original pick, there is a 33% chance that you pick the correct letter and a 67% chance that you do not. What this means is that 67% of the time the offer letter is one of the remaining letters. Once again, a rejection letter is ALWAYS pulled from the remaining letters. This means that 67% of the time the "other" letter will be the offer letter. Looking at it from the other direction, say you got lucky and picked the offer letter in your original guess. This only happens 33% of the time, so clearly you want to take the other letter.
The key to the problem is the fact that the rejection letter can not be removed from the stack of remaining letters.
Hope this helps.
CompBanker; I spent 30 minutes online trying to figure out this one and your answer is the epitome of clarity! Thank you very much.
You should switch because: At the beginning there is a 1/3 chance of picking the correct envelope and 2/3 chance of picking the incorrect envelope.
The 2/3 times that you pick the wrong envelope, the "doorman" is left with 2 envelops: a rejection and an acceptance. He must show you the sole remaining rejection and offer you to allow to swap. During these 2/3rds times, swapping will always win you that remaining acceptance letter.
The 1/3 times that you pick the correct envelope, the "doorman" is left with 2 envelopes: 2 rejections. He can show you either of his remaining rejections. If you swap in this scenario, you lose your acceptance for the rejection letter.
However, since a systematic always swap policy will win you the envelope 2/3rds of the time while always losing 1/3rds of the time and a systematic never swap policy only wins you the acceptance 1/3 of the time, you should always swap.
Finally: given 2 random envelopes with either an acceptance or rejection, it doesn't matter. However, you have more informaton than just that: the fact that you were shown a removed rejection and that there will always be a removed rejection.
No Simian, I'm actually not wrong. What you're saying is essentially the same thing. If the interviewer doesn't know what's in which envelope, then it DOES automatically introduce the possibility of him opening the job offer, since he DOES NOT KNOW WHERE IT IS, correct?
The concept thanks to which the probability doesn't change is that the interviewer ALWAYS removes the rejection letter, not the job offer. By removing one envelope, the remaining envelope essentially takes over the probability of the removed envelope, since we know now that the removed one is a rejection. Think of it this way, if the interviewer removed one envelope, but didn't open it or told us what's in it, would you still switch? There is no point in doing so because there are two envelopes and you don't know if they even have the job offer, so your chance has to be 50% for each of the options. No point in switching in this case. But if he removes a rejection letter, there is a 2/3 chance that the remaining envelope is an offer.
TH73 and Simian -- you two are arguing the same point.
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