I posted this on a forum I frequent, but so far haven’t gotten any answers I’m satisfied with.
An eccentric multi-billionaire with an interest in psychology and probability invites you to participate in a challenge.
He has taken a number of cards, and had a number printed on each of them. On the reverse side of the card, there is another number – twice the value of whatever is on the front.
The cards are put in a bag, and you are invited to draw one. You reach into the bag, feel around a bit, and pull out a card. You see that it shows 53. He will pay you that sum (in dollars), or allow you to pay a 10% premium (i.e. $5.30) to flip the card over and accept whatever sum is on the back, instead.
Do you pay to flip the card? Why/Why not?
* All rules are explained/guaranteed to you up front – the ‘flip’ offer was not in any way conditional upon whatever card/value you initially pulled, but rather was explained to you before you pulled the card.
* You have no particular idea as to the range of values printed on the cards. The experimenter is a billionaire with a rather cavalier attitude towards money. You will only be allowed to do this experiment once, so any knowledge you gain won’t be applicable to future tries.
* You can’t peek into the bag, feel the type on the back of the card, or any other ‘cheating’ solution like that. Looking at the card in front of you, you have no idea whether it was originally the ‘front’ or the ‘back’.
* The fact that the value you pulled ended in an odd digit is not in any way significant. The billionaire tells you that, subject to the range of the experiment that he had in mind, he had values randomly generated, down to fractions of a penny, then took the original value, and the doubled value, and rounded them to the nearest whole dollar (i.e. the ‘unrounded’ number on the reverse side could be anything from 26.5 to 26.999999, or 106 to 107.999999)
* The billionaire is honest, and, after the experiment, will allow you to fully examine everything (the cards, his number generator and it’s output, etc)
In answering why you did or didn’t pay, don’t rely on something simplistic like “I’m a gambler”, or “I’m risk averse”. Try to figure out mathematically/logically why one or the other choice yields the best expectation.
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