Jan submits this challenge for those of you up to the task.
You meet a man on the street who holds ten envelopes full of money. You get one of the envelopes. The rules are that you can look in the first one and accept or reject it, then look in the second one, and on and on. Once you reject an evelope, you cannot go back to it, it’s gone forever. You can only see later envelopes when you have rejected the current one. When you accept an envelope, the game is over.
According to this article, you should reject the first envelope and accept the next envelope you see that has more money than the first. It says:
According to Colley’s Rule, never accept the first offer, but instead note its qualities and accept the first subsequent offer which beats it. It was recently proved that this both increases the chances of making the best choice, while minimising the chances of making the worst.
In his book, Innumeracy, John Allen Paulos discusses this very game. He doesn’t mention Colley’s Rule and his results are different. Paulos proves that you should reject the first 37% of the choices and take the best one after that. Slightly different from just rejecting the first.
Your task, should you choose to accept it, is to create a simulation in Excel to see who, if either, is correct.