Subdivide Unconditional Probability Say, I have 6 urns, and balls of different color in each. I would like to compute the unconditional probability of drawing a red ball, when I somehow randomize over all urns. Now, my intuition tells me that $$ P(\text{drawing a red ball}) = \sum_{i=1}^6 P(\text{drawing a red ball | drawing from urn $i$} ) \cdot P(\text{drawing from urn $i$})$$ As I rely heavily on this and my math courses have been quite some time ago, I tried to prove it to be sure. I tried to use the Kolmogorov definition of conditional probability, but I didn't get it. Could someone give me some pointers?

Let $R$ be the event that a red ball is drawn. Let $U_i$ be the event that urn $i$ is used. Then
$$P(R) = P((R \cap U_1)\cup(R \cap U_2)\cup \cdots \cup (R \cap U_6)).$$ The events $(R \cap U_1), (R \cap U_2), \dots$ are mutually exclusive, so the probability of their union is the sum of their probabilities.
So $$P(R) = P \Big( \bigcup_{i=1}^6 (R \cap U_i) \Big) = \sum_{i = 1}^6 P(R \cap U_i) = \sum_{i = 1}^6 P(R \mid U_i) P(U_i),$$ as required.