Biased coin game 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.

Well, I've done a simulation where I used a simple criterion of saying the coin is biased if the current proportion is >0.55 else the coin is fair (middle between 0.5 and 0.6). Code in R:
res=replicate(1e3,{
  k=sample(c(0.5,0.6),1)
  s=sample(0:1,2e3,prob=c(1-k,k),replace=T)
  tmp=cumsum(s)/seq_along(s)
  ifelse(ifelse(k==0.6,1,0)==ifelse(tmp>0.55,1,0),1e4,-2e4)-seq_along(tmp)
})

and this is what the average reward plot looks like

note: i only tried up to 2000 draws as it has already converged somewhat by this time.
