Question

A strange submartingale? Suppose that a person writes two positive dollar amounts on two pieces of paper, one twice as large as the other, then seals the two pieces of paper into two separate envelopes. He lets you choose one of the envelopes at random and promises to pay you the dollar amount written inside. But before you open the envelope you’ve chosen, he always allows you to switch your choice. This is how you then reason: “If I have X dollars in my hand, the other envelope contains either X/2 or 2X, with equal probability, hence the expected amount there is 5X/4. So I should switch, and then use the same logic to switch again, and again, thus drive my expected amount to infinity.” This makes no sense, so point out the flaw in this reasoning.

Answer 1

lets assume the person wrote 10$ on one piece and 20$ (which is twice of 10$)

case 1: u choose piece with 10$ ,call it X which means second piece has 2X amount .since selecting a piece has probability of 1/2 its expected value is E(x)= 1/2*X+1/2*2X=3X/2=3*10/2=15$

case 2:u choose piece with 20$,call it X which means second piece has X/2 amount,again probability of selection is 1/2

its expected value is E(x)=1/2*X+1/2*X/2=3X/4=3*20/4=15$.

Here ,in both scenarios Expected value is same ,so whether u switch the packet or not expected value is same!!