As an example, say you're asked to solve for the following problem:
You have three bags, labeled A, B, and C. Bag A contains two red marbles and three blue marbles. Bag B contains five red marbles and one blue marble. Bag C contains three red marbles only. Suppose a bag is randomly selected (again, with equal probability), but you do not know which it is. You randomly draw a marble and observe that it is blue. What is the probability that the bag you selected this marble from is A?
I understand that you can calculate P(A|blue) using Baye's Theorem which would ask me to solve for P(blue|A) * P(A) / P(blue) which comes out to be 0.78. However, if I were to be asked this question outside the context of a class on bayesian statistics, I would think that the probability would be 0.75 or 3/4 since there are 4 blue marbles between the three bags. Since there is an equal chance of drawing from either bag, couldn't we disregard the probability of drawing from a given bag and focus on the distribution of blue marbles between bags? What is the error in my thinking here?
Suppose I pick a random marble from all the marbles, with each marble equally likely, and I show you that it's blue. In that case, your reasoning would make sense: you should figure that there's a 3/4 chance that my marble came from bag A.
However, in the scenario you described, the marbles are not equally likely to be chosen, because the bags have different numbers of marbles. Each blue marble in bag A has a higher probability of being chosen than the one blue marble in bag B has, because bag B has 6 marbles total while bag A only has 5 marbles. That's why the correct answer is a little bigger than 3/4.