You run into a magical genie. He lets you gamble once and only once, and gives you only the following cases to pick from:
>You have a 100% chance to earn $1
>You have a 10% chance to earn $10
>You have a 1% chance to earn $100
>You have a 0.1% chance to earn $1000
>You have a 0.01% chance to earn $10,000
>You have a 0.001% chance to earn $100,000
>You have a 0.0001% chance to win $1,000,000
>You have a 0.00001% chance to win $10,000,000
>You have a 0.000001% chance to win $100,000,000
>You have a 0.0000001% chance to win $1,000,000,000
These are the only available options. All of these have an expected pay out of 1 dollar. But clearly these are not equivalent value propositions. So how does one mathematically quantify the optimal choice and why?