Question: If there are 100 balls in urns A, B and C each (300 total), and the red balls in each urn are 45, 50 and 55, respectively, and the rest being yellow. If we pull a red ball from an urn, what is the probability that it was from urn A?
My attempt:
Let $U$ be the urn, and $b$ be the ball, then we are affectively asking $P(U=A|b=r)$. My first thought is to use Bayes, then we have:
$$P(U=A|b=r)=\frac{P(B=r|U=A)P(U=A)}{P(B=r)}$$
Since half the balls are red, the denominator is $\frac{1}{2}$. If the urn is $A$, then the probability of choosing the red ball is $\frac{45}{100}$ and the probability of randomly choosing urn A is $\frac{1}{3}$, then we have:
$$P(U=A|b=r)=\frac{\frac{45}{100}\frac{1}{3}}{\frac{1}{2}}=\frac{3}{10}$$
Your answer is correct. I'm thinking about the problem in a slightly different way. Each red ball is equally likely to be chosen. There are $150$ red balls in total, of which $45$ started in urn $A$. Therefore, the probability that your ball came from urn $A$ is $\frac{45}{150}=\frac{3}{10}$.