Can I Predict the Correct Dice Using Probability Distributions?

[aaaa202]

Say I have two dice which have different probability distributions but I don't know which probability distribution belongs to what dice.
I now throw one of them exactly one time and the result is x. I know that die A has probability pA(x) for rolling x and die B has probability pB(x).
I want to find out if the die I rolled was die A or B. Obviously my best strategy is to guess the die for which the probability for rolling x is the bigger. But what is my probability for being correct? I have been trying to figure out this problem for days. Obviously if pB(x)=0 while pA(x)>0 I have 100% chance of being correct. Moreover if pB(x)=pA(x) then I have 50% chance of being correct. But is there an expression for arbitrary probabilities and how do I find it?

 

[Dale]

Nice question. This would be a perfect application for Bayesian inference.

Are you familiar with Bayes theorem?

 

[aaaa202]

Yes P(XlY) = P(YlX) * P(Y)/P(X)

So in my case P(Alx) is the probability that the die is A given that it shows x, while P(xlA) is the probability that the die shows x given that it is die A and P(A) is the probability that it is die A given no prior information, i.e. P(A)=0.5 and P(x) is then the probability for tossing an x given no prior information, which would be 0.5*(pA(x)+pB(x)).. hmm.. is that correct?

 

[Dale]

Yes. I think that is correct.