Assume, there's a 50% chance I get a fair coin and 50% I get a biased coin with 0.6 chance of getting heads.

Then, I get to toss the coin I got as many times as I want, but each toss costs a dollar.

The goal of the game is to determine if the coin is fair or not. Let's say I win 10000 dollars for the correct guess and lose 20000 dollars for the wrong one.

Again, I can toss the coin I got as many times as I want, but it's clear I don't want to toss it more than 10k times because then my profit is 0.

So, should I play the game? And what's the best number of tosses to maximize the profit/minimize loss?

What I've done so far, is found a Neyman-Pearson test statistic to see if the coin is biased:

$$T = \frac{n\_0-n\_1}{\sqrt{n\_0+n\_1}}$$

So, if I set the false alarm rate and missed signal rate I can find the number of tosses. But I'm not sure how to correctly set them.