Suppose we have two fair (faces are equally likely) dice d$1$ and d$2$. d$1$ has $4$ faces of $4$ and $2$ faces of $2$ while d$2$ has $3$ faces of $3$ and $3$ faces of $5$.
We roll both dice. The one with larger value wins. The question is what's the possibility d1 wins. It seems like a simple question and we can jump to the answer real quick $(4/6)\cdot(3/6) = 1/3$. But I'm more curious about the reasoning behind the quick solution. So I'm trying to be a bit rigorous and here comes my analysis.
Since this is a probability problem, in order to solve it we need to have a probabilistic model (accomplished by step$1$ and step$2$) and then apply probability axioms to get the answer (step$3$).
- we'll first need to find out the sample space, in this case it is $\{(4,3), (4,5), (2,3), (2,5)\}$.
- The next step is to have a probability laws i.e. a function $P$ assigning every single element subset of the sample space to a real number.
- And finally by the additivity axiom, we have the answer = $P({(4,3)})$
Here's the real question. How to derive the function $P$ in step$2$ and by which axioms or properties?
I thought about using the conditional probability, i.e $P(d1=4 \text{ AND } d2=3) = P(d1=4 \vert d2=3) \cdot P(d2=3)$, but the problem here is that the sample spaces are different for d1 and d2. And to my knowledge the formula only works in the same sample space.
Hope I made myself clear. Please let me know what you think.