Suppose I present you with two boxes. I tell you that each box has a positive amount of money, and that one amount is double the other. I allow you to choose a box.
After you open it and determine how much is inside, I offer to let you switch to the other box in exchange for 10% of the amount in the first box. E.g., if your box has $10, I offer to sell you the other box for $1.
Should you take the deal, or should you keep your original box?
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5 comments:
Continuing with your example, if I keep what I picked, I will get $10. If I switch, I will either spend a dollar to win $20 (for a net win of $19) or spend a dollar to win $5 (for a net win of $4).
The two outcomes have equal probability, and the average result is $23/2 or $11.50, so I'm more likely to come out ahead by switching.
Care to play this game one million times? I would need to add the restriction that you pay $n to play, where n is the lower value of the 2. If you really do benefit by switching, it should be to your advantage.
Oops, I meant $3n/2
In fact, it's better not to pay the premium to switch - in the long run it makes no difference whether you switch or not.
The correct expected return calculation is left as an exercise for the reader.
I can't believe I typed the words "I'm more likely to come out ahead by switching."
After encountering a similar problem, I discovered that this belongs to a family of paradoxes that has generated "a voluminous literature" with new papers being published every year.
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