switching job offer envelopes puzzle (a.k.a. Monty Hall problem)

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Puzzle:
Three envelopes are presented in front of you by an interviewer. One contains a job offer, the other two contain rejection letters. You pick one of the envelopes. The interviewer then shows you the contents of one of the other envelopes, which is a rejection letter. The interviewer now gives you the opportunity to switch envelope choices. Should you switch?

Common solution:
The answer is yes. Say your original pick was envelope A. Originally, you had a 1/3 chance that envelope A contained the offer letter. There was a 2/3 chance that the offer letter was either in envelope B or C. If you stick with envelope A, you still have the same 1/3 chance. Now, the interviewer eliminated one of the envelopes (say, envelope B), which contained a rejection letter. So, by switching to envelope C, you now have a 2/3 chance of getting the offer and you’ve doubled your chances.

Discussion:
One critical piece of information is missing, and the answer actually depends on it. Does the interviewer know the content of her envelopes, or opens one at random?

  • If the the interviewer knows the content of her envelopes, then the common solution is correct. This is related to the so called Standard assumptions (wikipedia article on the Monty Hall problem)
  • If the the interviewer does not know the content of her envelopes, then the common solution is not correct. In this case we are dealing with conditional probabilities, given that the interviewer at random eliminated one of the rejection envelopes. This gives you a 50% conditional probability of success, and switching envelopes will not increase your chances.To make it more intuitive, consider the “law of large numbers” arguments. Suppose you play this random game 9 million times. Approximately 3 million times the interviewer at random will eliminate the offer letter. Out of the remaining approximately 6 million times, you will choose an offer letter about 3 million times. Your chance of success will not depend on switching envelopes (which is a great example on conditional probabilities).

This discussion is related to the Monty Hall problem: The probability puzzle that makes your head melt (by the BBC)