Given two events, A and B, to “find the probability of A or B” means to find the probability that either event A or event B occurs.
We typically write this probability in one of two ways:
- P(A or B) – Written form
- P(A∪B) – Notation form
The way we calculate this probability depends on whether or not events A and B are mutually exclusive or not. Two events are mutually exclusive if they cannot occur at the same time.
If A and B are mutually exclusive, then the formula we use to calculate P(A∪B) is:
Mutually Exclusive Events: P(A∪B) = P(A) + P(B)
If A and B are not mutually exclusive, then the formula we use to calculate P(A∪B) is:
Not Mutually Exclusive Events: P(A∪B) = P(A) + P(B) - P(A∩B)
Note that P(A∩B) is the probability that event A and event B both occur.
The following examples show how to use these formulas in practice.
Examples: P(A∪B) for Mutually Exclusive Events
Example 1: What is the probability of rolling a dice and getting either a 2 or a 5?
Solution: If we define event A as getting a 2 and event B as getting a 5, then these two events are mutually exclusive because we can’t roll a 2 and a 5 at the same time. Thus, the probability that we roll either a 2 or a 5 is calculated as:
P(A∪B) = (1/6) + (1/6) = 2/6 = 1/3.
Example 2: Suppose an urn contains 3 red balls, 2 green balls, and 5 yellow balls. If we randomly select one ball, what is the probability of selecting either a red or green ball?
Solution: If we define event A as selecting a red ball and event B as selecting a green ball, then these two events are mutually exclusive because we can’t select a ball that is both red and green. Thus, the probability that we select either a red or green ball is calculated as:
P(A∪B) = (3/10) + (2/10) = 5/10 = 1/2.
Examples: P(A∪B) for Not Mutually Exclusive Events
The following examples show how to calculate P(A∪B) when A and B are not mutually exclusive events.
Example 1: If we randomly select a card from a standard 52-card deck, what is the probability of choosing either a Spade or a Queen?
Solution: In this example, it’s possible to choose a card that is both a Spade and a Queen, thus these two events are not mutually exclusive.
If we let event A be the event of choosing a Spade and event B be the event of choosing a Queen, then we have the following probabilities:
- P(A) = 13/52
- P(B) = 4/52
- P(A∩B) = 1/52
Thus, the probability of choosing either a Spade or a Queen is calculated as:
P(A∪B) = P(A) + P(B) – P(A∩B) = (13/52) + (4/52) – (1/52) = 16/52 = 4/13.
Example 2: If we roll a dice, what is the probability that it lands on a number greater than 3 or an even number?
Solution: In this example, it’s possible for the dice to land on a number that is both greater than 3 and even, thus these two events are not mutually exclusive.
If we let event A be the event of rolling a number greater than 3 and event B be the event of rolling an even number, then we have the following probabilities:
- P(A) = 3/6
- P(B) = 3/6
- P(A∩B) = 2/6
Thus, the probability that the dice lands on a number greater than 3 or an even number is calculated as:
P(A∪B) = P(A) + P(B) – P(A∩B) = (3/6) + (3/6) – (2/6) = 4/6 = 2/3.