How to Find the Probability of A or B (With Examples)


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(AB) 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.

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