Here is another one, for those who know about probability:

In an imaginary TV show, there is a large horizontal clock face with an hour hand mounted on a ball bearing. The hand is pushed so that it rotates many times. When at last it comes to a standstill, its position in terms of hours past midnight (larger than zero, up to 12) is measured exactly, including the fractional part. The "candidate" wins one dollar divided by that position number: 1/12 dollar for 12 o'clock, 1/6 dollar for 6 o'clock, 60 dollars for 1 minute past midnight, etc. Of course, the amount is rounded to full cents. The TV company asks an insurance company to take the risk. What would be a fair insurance premium? Or, in other words: how much would it be worth to be that candidate (strictly by face value, disregarding the usefulness for a particular person)?

(Courtesy of a friend of mine who minored in maths.)

The connection to the OP: Assume the TV company operates this oracle to determine an amount N, invisible for you, but closely watched by an impartial jury. Then the company writes one check for N and another one for twice N. The rest is as in the OP.