Solution to the Deal or No Deal Problem (no statistical data present)
So I've been brainstorming kind of a model that could be used to estimate a range of possible banker's offers given an expected value of x.
First, we collect sufficient data on the history of banker's deals and expected values. We then calculate the ratio of each offer to its corresponding "expected value" cell. We write those values down.
Then we write 1) the mean of the ratios, 2) the average of all ratios below the mean, and 3) the average of all ratios above the mean.
Then #2 and #3 indicate the estimated mimimum offer-expected value ratio and the estimated maximum offer-expected value ratio, respectively.
I have no idea how to calculate the probability of getting it right, but it is without doubt greater than the estimated range / (the estimated maximum amount of money to potentially have lost - estimated minimum amount of money to potentially have lost)