Anyone have Two Envelopes?
Your friend has hit the lottery and, in an act of generosity, decided to give you some money. He chooses a random amount of Rupees , where is a real number between 1,000 and 1,000,000.
He then places Rs. in one envelope, Rs in another and places them both on a table.He then offers you one of them to keep.
You pick one of them up and you find Rs inside.
Now, your friend offers you the opportunity to change your mind and choose the other envelope. But if you change your mind and choose the other envelope, you will have to stick with it.
Given these circumstances, which option has the highest expected value?
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For simplicity, let A and B be the envelopes with X and 2 X Rupees respectively. And let R s 3 1 4 1 6 be Y .
If we chose A initially, that means X = Y and if we choose the other envelope, we will get B which has 2 X = 2 Y .
If we chose B initially, that means 2 X = Y and if we choose the other envelope, we will get A which has X = Y / 2 .
Now, there is a 1 / 2 we chose A initially and a 1 / 2 chance we chose B initially.
Therefore, the expected gain from changing is 2 1 ( 2 Y ) + 2 1 ( 2 Y ) = 4 5 Y which is greater than the Y we would have gotten if we had stuck with the original envelope.