Given two events, A and B, to “find the probability of neither A nor B” means to find the probability that **neither event A nor event B occurs.**

We use the following formula to calculate this probability:

**P(Neither A Nor B) = 1 – ( P(A) + P(B) – P(A∩B) )**

where:

- P(A): The probability that event A occurs.
- P(B): The probability that event B occurs.
- P(A∩B): The probability that event A and event B both occur.

The following examples show how to use this formula in practice.

**Example 1: Probability of Neither A Nor B (Basketball Players)**

Suppose the probability that a given college basketball player gets drafted into the NBA is **0.03**.

Also suppose the probability that a given college basketball player has a 4.0 GPA is **0.25**.

Also suppose the probability that a given college basketball player has a 4.0 GPA *and* gets drafted into the NBA is **0.005**.

If we randomly select some college basketball player, what is the probability that they neither get drafted nor have a 4.0 GPA?

**Solution**:

- P(drafted) = 0.03
- P(4.0 GPA) = 0.25
- P(drafted ∩ 4.0 GPA) = 0.005

Thus, we can calculate:

- P(Neither drafted Nor 4.0 GPA) = 1 – ( P(drafted) + P(4.0 GPA) – P(drafted ∩ 4.0 GPA) )
- P(Neither drafted Nor 4.0 GPA) = 1 -(.03 + .25 – .005)
- P(Neither drafted Nor 4.0 GPA) = 0.715

If we randomly select some college basketball player, the probability that they neither get drafted nor have a 4.0 GPA is 0.715 or **71.5%**.

**Example 2: Probability of Neither A Nor B (Exam Scores)**

Suppose the probability that a given student receives a perfect score on a final exam is **0.13**.

Also suppose the probability that a given student used a new studying method is **0.35**.

Also suppose the probability that a given student received a perfect score *and* used a new studying method is **0.04**.

If we randomly select some student, what is the probability that they neither received a perfect score nor used a new studying method?

**Solution**:

- P(perfect score) = 0.13
- P(new method) = 0.35
- P(perfect score ∩ new method) = 0.04

Thus, we can calculate:

- P(Neither perfect score Nor new method) = 1 – ( P(perfect score) + P(new method) – P(perfect score ∩ new method) )
- P(Neither perfect score Nor new method) = 1 – (0.13 + 0.35 – 0.04)
- P(Neither perfect score Nor new method) = 0.56

If we randomly select some student, the probability that they neither received a perfect score nor used a new studying method is 0.56 or **56%**.

**Additional Resources**

The following tutorials explain how to perform other calculations related to probabilities:

How to Find the Probability of A or B

How to Find the Probability of A and B

How to Find the Probability of A Given B

How to Find the Probability of “At Least One” Success