This is a great probability question that arose from the famous necktie paradox. Have fun!
Note: If you could all please input your answers using the previously noted html code it will allow others to answer without having a spoiler. Also, in your answer, do not include any ' or " or it will not work properly.
Question
There are two envelopes in front of you each with a non-zero sum of money. You are informed that one has twice as much money as the other. You are then allowed to select either envelope and keep the money inside. After you select one, but before opening it, you are given the option to change your mind and switch to the other one. You think to yourself that if your envelope has x dollars there is a 50% chance the other one has x/2 dollars and a 50% chance it has 2x dollars. The expected return, you compute, is .5[.5x + 2x]=1.25x which seems like a favorable gamble. Do you switch, and if so, why? Assume you will take any good gamble and avoid any bad one.
Good Luck!
It doesnt matter whether you switch the envelope or not - they both have the same expected value. Suppose one envelope has X dollars, and the other one has 2X dollars. The expected payout of your envelope is 0.5[X + 2X] = 1.5X. The expected payout of switching is 0.5[X + 2X] = 1.5X (50% chance of starting with the lower one, and 50% chance of starting with the higher one).
I believe that your "expected payout of switching" equation is incorrect, as you will have either .5[.5X + 2x]. Also, not to be a douche, but I would recommend waiting a few hours on Tuesday if you you cannot input the form correctly, seeing your answer right below the question is not all that cool. Can be forgiven though.
You're incorrect. ".5(.5x + 2x)" would be the formula if the payout of the second envelope was four times the sum of the first. You could use either .5x + x or x + 2x..... Regardless of which way you compute it, your expected sums would be the same which makes sense since there is nothing to change your expected return from your first choice.
The only way in which probability of an outcome would chance is if an event occured after making your first selection such as what occurs in the Monty Hall problem.
I dont think that switching makes a difference. Assume that switching would offer you a higher expected return and therefore you make the switch. If you were now presented with the opportunity to switch back then the analysis would be the same and you would switch back again. Now we have concluded that envelope A has a higher return than envelope B and B has a higher return than A, which is a contradiction. The only conclusion is that switching does not make a difference to begin with
I have no idea if this makes sense or not but my rationale
whoops, double post
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