Mutually Exclusive vs Independent Events

People often confuse the concept of mutually exclusive events with independent events. In fact, these are two different things.

Let A and B be any two events associated with a random experiment E. P(A) is called the “Probability of A”. Similarly, we can define probability of B as P(B), probability of A or B as P(A∪B), and probability of A and B as P(A∩B). Then,  P(A∪B)=P(A)+ P(B)-P(A∩B).

However, two events said to be mutually exclusive if the occurrence of one event does not affect the other. In other words, they cannot occur simultaneously. Therefore, if two events A and B are mutually exclusive then A∩B=∅ and hence, that implies P(A∪B)=P(A)+ P(B).

Let A and B be two events in a sample space S. Conditional probability of A, given that B has occurred, is denoted by P(A | B) and is defined as; P(A | B)=P(A∩B)/P(B), provided P(B)>0. (otherwise, it is not defined.)

An event A is said to be independent of an event B, if the probability that A occurs is not influenced by whether B has occurred or not. In other words, the outcome of the event B has no effect on the outcome of the event A. Therefore, P(A | B)= P(A). Similarly, B is independent of A if P(B) = P(B | A). Hence, we can conclude that if A and B are independent events, then P(A∩B)=P(A).P(B)

Assume that a numbered cube is rolled and a fair coin is flipped. Let A be the event that obtaining a head and B be the event that rolling an even number. Then we can conclude that events A and B are independent, because that outcome of one does not affect the outcome of the other. Therefore, P(A∩B)=P(A).P(B)=(1/2)(1/2)=1/4. Since P(A∩B)≠0, A and B cannot be mutually exclusive.

Suppose that an urn contains 7 white marbles and 8 black marbles. Define event A as drawing a white marble and event B as drawing a black marble. Assuming each marble will be replaced after noting down its color, then P(A) and P(B) will always be the same, no matter how many times we draw from the urn. Replacing the marbles means that the probabilities don’t change from draw to draw, no matter what color we picked on the last draw. Therefore, event A and B are independent.

However, if marbles were drawn without replacement, then everything changes. Under this assumption, the events A and B are are not independent. Drawing a white marble the first time changes the probabilities for drawing a black marble on the second draw and so on. In other words, each draw has an effect on the next draw, and so the individual draws are not independent.

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