Problem: Given a random distribution A and random variable X~A, X$\in${1,2,3}, with f.x. probabilities $P_{\textrm{A}}(X = 1) = P_{\textrm{A}}(X = 2) = 0.1$ and $P_{\textrm{A}}(X = 3) = 0.8$. Let's say I'm sampling from this distribution, and also throwing a fair (non-biased) 3-sided dice with numbers 1, 2 and 3. What is the probability of the number on the dice and the sampling from the distribution agree?
My solution From intuition I would say that the probability is:
$P = \sum_{i=1}^3 P_{\textrm{Dice}}(i) * P_{\textrm{A}}(X = i)$
The problem
This gives us $P = \sum_{i=1}^3 \frac{1}{3} * P_{A}(X = i) = \frac{1}{3} \sum_{i=1}^3 P_{A}(X = i) = \frac{1}{3} 1 = \frac{1}{3}$
So the distribution we are sampling from doesn't matter..., the answer is always 1/3. I guess that might be ok, but I'm not so sure. Am I on the right track? :)