Why are two draws with replacement from an urn independent?
Swiss commemorative stamp of mathematician Jakob Bernoulli, issued 1994, displaying the formula and the graph for the law of large numbers, first proved by Bernoulli in 1713.
In probability theory, two events are considered independent if the occurrence of one does not change the probability of the other.1
When drawing two balls from an urn with replacement, the events are independent because the composition of the urn remains identical for both draws. By returning the first ball to the urn and thoroughly mixing the contents before the second draw, the probability of selecting a specific color remains constant for every trial.1
In contrast, drawing without replacement is dependent because the first draw alters the total number of balls and the count of the specific color remaining in the urn, thereby changing the conditional probability for the second draw. Because replacement ensures the state of the system is reset, the outcome of the first draw provides no information that would adjust the likelihood of the second, satisfying the formal definition of independence.21