Question

Jamal picks a marble at random (3 green, 3 orange, 1 pink). Without putting the first marble back, he picks a second marble at random.

Are these two events dependent or independent?

Answer 1

Dependent. Because Jamal does not replace the first marble, the first draw changes the composition of the urn and thus changes the probabilities for the second draw.

For example, P(second is green) overall = 3/7, but
P(second is green | first was green) = 2/6 = 1/3, and 1/3 ≠ 3/7,
so the conditional probability differs from the unconditional one — the events are dependent.