Addition Rules in Probability

Addition rules are important in probability. These rules provide us with a way to calculate the probability of the event "A or B," provided that we know the probability of A and the probability of B. Sometimes the "or" is replaced by U, the symbol from set theory that denotes the union of two sets. The precise addition rule to use is dependent upon whether event A and event B are mutually exclusive or not.

Addition Rule for Mutually Exclusive Events

If events A and B are mutually exclusive, then the probability of A or B is the sum of the probability of A and the probability of B.  We write this compactly as follows:

P(A or B) = P(A) + P(B)

Generalized Addition Rule for Any Two Events

The above formula can be generalized for situations where events may not necessarily be mutually exclusive. For any two events A and B, the probability of A or B is the sum of the probability of A and the probability of B minus the shared probability of both A and B:

P(A or B) = P(A) + P(B) - P(A and B)

Sometimes the word "and" is replaced by ∩, which is the symbol from set theory that denotes the intersection of two sets.

The addition rule for mutually exclusive events is really a special case of the generalized rule. This is because if A and B are mutually exclusive, then the probability of both A and B is zero.

Example #1

We will see examples of how to use these addition rules. Suppose that we draw a card from a well-shuffled standard deck of cards. We want to determine the probability that the card drawn is a two or a face card. The event "a face card is drawn" is mutually exclusive with the event "a two is drawn," so we will simply need to add the probabilities of these two events together.

There are a total of 12 face cards, and so the probability of drawing a face card is 12/52. There are four twos in the deck, and so the probability of drawing a two is 4/52. This means that the probability of drawing a two or a face card is 12/52 + 4/52 = 16/52.

Example #2

Now suppose that we draw a card from a well-shuffled standard deck of cards. Now we want to determine the probability of drawing a red card or an ace. In this case, the two events are not mutually exclusive. The ace of hearts and the ace of diamonds are elements of the set of red cards and the set of aces.

We consider three probabilities and then combine them using the generalized addition rule:

• The probability of drawing a red card is 26/52
• The probability of drawing an ace is 4/52
• The probability of drawing a red card and an ace is 2/52

This means that the probability of drawing a red card or an ace is 26/52+4/52 - 2/52 = 28/52.