I've been trying to figure this problem out.
We are attempting to determine in coins are fair or unfair by observing the results of a series of coin flips. Each new person has a 50% chance of getting an unfair coin. Unfair coins have a 75% chance of landing heads. We may request any number of coin flips from the person, but the coin flips are deducted from our flip total. Assuming that we get 15 added to our flip total when we guess correctly and 30 subtracted from our flip total when we guess incorrectly, what's a good strategy for gaining the most flips over a series of games?
I've been attempting to formulate a good mixed strategy when playing a series of these games, but I'm out of my depth. Help would be much appreciated.
Assuming you set the number of throws before throwing coins.
For n=1, 2, 3, and for 0 <= k <= n, calculate the probability of throwing k heads in n throws, once for fair coin, once for an unfair coin.
From this you figure whether you would guess “fair” or “unfair” after k heads in n throws, and what is the probability of having k heads and a correct guess. Add for all k, and you have the probability p(n) of making a correct guess after n throws.
You can expect 15 p(n) - 30 (1-p(n)) - n points if you guess after n throws. You calculate this for n = 0 to 45 and pick the best n.
Assuming you can decide after any number of throws whether to guess or to do another throw, your expected outcome should be better, but the strategy is hard. BTW. I don’t think a mixed strategy would help.