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My favourite probability problem

Posted By: joe freedman
Date: Friday, 28 August 2009, at 2:56 p.m.

In Response To: My favourite probability problem (Stein Kulseth)

There seems to be something missing here. First, there's no indication of the probability that the envelope will contain the check. Is it 50%? Also, there's no indication that choosing to accept the cash would preclude you from also taking the envelope.

Obviously, if taking the $1000 does not stop you from also taking the envelope, there is no reason not to take the money and then the envelope. If, however, you cannot take both (and that would seem to be the assumption -- why would you otherwise "plainly turn down" the cash -- then the answer depends on the probability of the envelope containing a check. Putting aside what the Predictor said, if that probability exceeds 1/1000, you should decline the cash and take the envelope.

If you have no way of knowing the probability, then it's a matter of taste -- either a sure $1000 or some percentage equity in a million. Let's assume that only a gambler, or a wealthy person to whom $1000 means nothing, would forego the sure cash.

Now, consider what the predictor said -- that you would plainly turn down the envelope. That means either that (1) the odds of the check being there are known to exceed .1% or (2) the odds are unknown (or exactly 1%) AND you are a gambler (or very wealthy), and the predictor knows it. If either either of those are the case, take the envelope. If not -- i.e., if the odds are <.1% or they are unknown or = .1% and you are not a gambler (or wealthy), take the cash.

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