At each step it’s pretty easy to calculate the probability that the patient has each disease, and the probability that the patient will die if untreated.
You can get a .5 chance of saving the patient if you simply choose to treat for one disease and stick with that treatment. Let me arbitrarily say that if I choose this option I will treat for fungus.
I think a simple one move look ahead can give the optimum strategy: compare the chance of the patient dying on this turn to the chance of waiting saving the patient’s life. It doesn’t do any good to wait for more evidence if the extra evidence won’t change your treatment anyway. So the chance of waiting saving patient is the confidence that you will have correctly identified the disease after waiting an extra turn minus .5 if changes your treatment and zero otherwise.
For example, if I have 1 red and one black card already and have not yet treated the patient and 4 turns have gone by, the chance of the patient dying next turn is I think 4⁄36. The the chance of waiting saving the patient’s life is .5 .5 (x - .5) where the number x which I am too lazy to calculate is the probability the patient has an allergy given two black cards and one red one. (The first .5 is because there’s only a coin flip chance of getting test results, the second because if I draw a red card it won’t change my mind how to treat the patient).
I’d like to code your proposed strategy up and measure it by simulations, but I can’t quite figure out what it is. Does your strategy start with default-treating someone for fungus, or with waiting? Does it switch to treating an allergy on the first suggestive evidence for allergy, or does it wait for conclusive proof?
At each step it’s pretty easy to calculate the probability that the patient has each disease, and the probability that the patient will die if untreated.
You can get a .5 chance of saving the patient if you simply choose to treat for one disease and stick with that treatment. Let me arbitrarily say that if I choose this option I will treat for fungus.
I think a simple one move look ahead can give the optimum strategy: compare the chance of the patient dying on this turn to the chance of waiting saving the patient’s life. It doesn’t do any good to wait for more evidence if the extra evidence won’t change your treatment anyway. So the chance of waiting saving patient is the confidence that you will have correctly identified the disease after waiting an extra turn minus .5 if changes your treatment and zero otherwise.
For example, if I have 1 red and one black card already and have not yet treated the patient and 4 turns have gone by, the chance of the patient dying next turn is I think 4⁄36. The the chance of waiting saving the patient’s life is .5 .5 (x - .5) where the number x which I am too lazy to calculate is the probability the patient has an allergy given two black cards and one red one. (The first .5 is because there’s only a coin flip chance of getting test results, the second because if I draw a red card it won’t change my mind how to treat the patient).
I’d like to code your proposed strategy up and measure it by simulations, but I can’t quite figure out what it is. Does your strategy start with default-treating someone for fungus, or with waiting? Does it switch to treating an allergy on the first suggestive evidence for allergy, or does it wait for conclusive proof?