Probability - urns

Suppose we have three urns with black and white balls distributed as follows:

Urn A has $$10$$ black balls and $$20$$ white balls Urn B has $$10$$ black balls and $$10$$ white balls Urn C has $$10$$ black balls and $$1$$ white ball. Suppose we choose an urn (uniformly at random) and draw a ball (uniformly at random) from that urn. What is the probability that the ball is white?

I believe the answer is $$\frac{1}{3} \cdot \frac{2}{3} + \frac{1}{3} \cdot \frac{1}{2} + \frac{1}{3} \cdot \frac{1}{11}$$ Is that correct?

• Yes, your reasoning is correct!
– Remy
Oct 12, 2018 at 9:59
• Yes, the answer is correct. Oct 12, 2018 at 10:00

Application of law of total probability is what they call this: $$P(E)=P(A)P(E\mid A)+P(B)P(E\mid B)+P(C)P(E\mid C)$$ where $$A,B,C$$ are mutually exclusive and covering.