The Puzzle:
You have three bags, each containing two marbles.
Bag A contains two white marbles,
Bag B contains two black marbles,
and Bag C contains one white marble and one black marble.
You pick a random bag and take out one marble.
It is a white marble.
What is the probability that the remaining marble from the same bag is also white?
The Solution:
2/3 (not 1/2)
I am having a hard time wrapping my head around this, I still do not believe that the case where you count choosing bag A as 2 separate cases is valid.
Perhaps slightly rephrasing the question will help.
Instead of asking what is the probability that the remaining marble from the same bag is also white, we may equivalently ask, if we draw a white marble, what is the probability that we drew the marble from bag A? From this slightly different phrasing of the question we arrive at the solution of $\frac{2}{3}$ since $2$ of the $3$ white marbles are in bag A.
Equivalence with the Monty Hall Problem
In the Monty Hall problem, one might think that once the first door is opened the probability of choosing the right door becomes $\frac{1}{2}$ because there are only 2 unopened doors left, 1 with a goat and 1 with the car. Much like one might think that the probability the other marble is white in this problem is $\frac{1}{2}$ since there are white marbles in only two bags, and in one bag the marble's partner is black and the in the other it is white.
But to do so in both cases is a mistake because it forgets the original probability that the door you pick has a goat behind it or that the white marble comes from bag A. That is, when you first pick a door, the probability that you pick a door with a goat behind it is $\frac{2}{3}$. The key is to realize that this does not change when one of the other doors is opened. The probability that the door you picked has a goat behind it is still $\frac{2}{3}$. In the same way, the probability of drawing a white marble from bag A is $\frac{2}{3}$ before it is drawn, it is still $\frac{2}{3}$ after it has been drawn, and this implies that the probability the remaining marble is white is also $\frac{2}{3}$.
Here is Sal Khan's exposition of the Monty Hall problem https://www.youtube.com/watch?v=Xp6V_lO1ZKA.