Let us consider such a scenario: Box A contains one green ball and one red ball, Box B contains two green balls and one red ball. Assume also that we can only choose one box each time, and the probability of choosing Box A is 1/3, the probability of choosing Box B is 2/3. Then the probability of obtaining one red ball in this case is
$$P(R)=P(A)P(R|A)+P(B)P(R|B)=\frac{1}{3}\cdot\frac{1}{2}+\frac{2}{3}\cdot\frac{1}{3}$$ On the other hand, a probability $E$ can be defined as $$P(E)=\frac{\mbox{the number of outcomes of E}}{\mbox{the size of the sample space}}$$ But what is the sample space in this case? I tried something like $$S=\{(A,G),(A,R),(B,G_1),(B,G_2),(B,R')\}$$ This apparently doesn't work for two reasons:
- The probability of choosing A and B are not equal.
- Choosing Box A and choosing Box B is repelling to each other, so we should not have $(A,*)$ and $(B,*)$ occur simultaneously.
But then I have no idea.
No, probability of event can be defined as fraction of number of outcomes in it only if this outcomes have equal probabilities. In general case, you can't even make such probability space - for example, if one of outcomes has irrational probability.