Imagine you have two buckets filled with marbles. Bucket #1 contains 30 red marbles and 10 black marbles, while Bucket #2 holds 20 red and 20 black marbles. If a friend secretly picks a marble from one of these buckets and reveals it's red, what is the likelihood it was drawn from Bucket #1?
Now, consider that your friend replaces the marble and proceeds to draw two marbles, one at a time, from the same bucket. After drawing the first marble and putting it back, she draws a second one. Both marbles are red. What is the probability that these marbles were from Bucket #1?