Consider a lottery where you have a 1 in 2 chance of winning $2, a 1 in 4 chance of winning $4, a 1 in 8 chance of winning $8, and so on. The expected value of this is $(2/2 + 4/4 + 8/8 + …), which is infinite. But obviously you wouldn’t buy these tickets if they cost $50 each. This is because you’d probably run out of money before getting to the long run that the expected value measures.
I’m thinking of a natural number. What distribution do you put on this natural number? In particular, what is its median, or even its mode?
You’re playing a game where you have a 40% chance to win your bet and a 60% chance to lose, but you employ the Martingale strategy, where you double your bet each time? The only states where you stop betting are when you win 1, so your expected value is +1. However, after n bets, your expected value is 1-(6/5)^n, which diverges to negative infinity as n approaches infinity.
You get to have a 1 in X chance for $X, but you get to pick any value for X in the range [1, inf). What value do you pick? Why specifically that value?
Some guy runs up to you in the street, and he’s like “You should believe in my god Zulu, who will grant you his wealth if you believe in him.”
You: “I think that’s very unlikely. I think that only has like a 1 in [X] chance of being true.”
Guy: “Well, lucky for you, the amount of wealth that he’ll give you is $[X],000. So, your expected value for believing is $1,000.”
By the way, the number I was thinking of earlier was 9.
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