I’m thinking of two numbers between 0 and 1. Your goal is to guess a number which falls in between my two numbers. Each guess costs you $1, and if you guess correctly you win the reciprocal of the length of my range (ie if I am thinking of 0.2 and 0.4, a correct guess wins you $5). At any time you may request that I choose of a new pair of numbers, and of course I will pick a new pair of numbers whenever you win.
What’s your strategy? Under certain conditions, this game is fair. How might you be able to have a positive expectation?