Expected Return Question

Hi!
I am wondering what is the expected return on earning an infinite sum of money which has an almost zero chance of occurrence - probability of, say, one in a trillion (but NOT zero). Will the expected return be still infinite?
Thanks

 

Dig out your Introduction To Calculus book and reacquaint yourself with L^Hopital's Rule and other "limit as x approaches infinity" products and quotients.

 

E = p x W - (1-p)L

where: p = probability of win ~ 0
W = infinity
L = 0

plug in and you get: E = p x W

If p is small but still finite , then E = infinity. As p goes to zero in the limit, E becomes an undefined quantity.

This means it can be either 0 or infinity, as already assumed, or any other value in between depending on many other factors.

The question is: when does E start becoming an undefined quantity?

This is related to St Petersburg paradox. The paradox arises because of the use of infinity in a decision process.

No natural decision process can involve infinity since the universe itself is not infinite.

Drop infinity and you are ok. In this game your ticket will cost a lot of money.

 

thanks..but i do not understand why you think p the should go down to zero....it is a fairy small number but it is a fixed one..and infinity is also a "fixed" value - there is no need to approach it though a limit...

 

The simple question you should ask yourself is this -- is a fraction of infinity less than infinity?

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<i> That is whole, this is whole
From the whole, the whole is subtracted
When the whole is taken from the whole
The whole still will remain —
--Upanishad. </i>