The Addition Rule of Probability

Probabilities of an Event or Another Event

Probability tells us how likely (how probable) it is an event will happen.

For example, it tells us that when a die is rolled, the probability of rolling a 1 is 16.

It tells us that when a die is rolled, the probability of rolling a 6 is 16.

Imagine we wanted to find the probability of rolling a 1 or rolling a 6.

The Addition Rule

To find a probability of one event or another event...

Probability of 1 or Probability of 6

...replace the or with a +...

Probability of 1 + Probability of 6

This is the addition rule.

How to Find the Probability of an Event or Another Event

Question

What is the probability of rolling a 1 or rolling a 6 on a die?

Step-by-Step:

1

Write down what we are trying to find out.

Probability of rolling a 1 or Probability of rolling a 6

2

Replace or with +.

Probability of rolling a 1 + Probability of rolling a 6

3

Find the probability of rolling a 1.

The probability of rolling a 1 is 16.

4

Find the probability of rolling a 6.

The probability of rolling a 6 is 16.

5

Substitute the probability of rolling a 1 (16) and rolling a 6 (16) into the formula.

16 + 16 = 26

Answer:

The probability of rolling a 1 and rolling a 6 is 26.

A Formula for the Addition Rule of Probability

The formula for finding the probability of event A or event B is shown below:

Let's use the formula in an example.

Question

A spinner will spin and come to rest pointing at a color.

What is the probability of the spinner pointing at red or blue?

Step-by-Step:

1

Start with the formula.

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

2

Define the events in our example.

  • Let R be the event of the spinner coming to rest at Red. P(R) is the probability of spinning Red.

  • Let B be the event of the spinner coming to rest at Blue. P(B) is the probability of spinning Blue.

We can rewrite the addition rule:

P(R and B) = P(R) + P(B)

3

Find the probability of the spinner coming to rest at Red.

There are 2 ways of the spinner stopping at Red...

...out of 8 equally probable outomes.

The probability of the spinner stopping at Red is 28. P(R) = 28.

4

Find the probability of the spinner stopping at Blue.

There are 3 ways of the spinner stopping at Blue our of 8 equally probably outcomes.

The probability of the spinner stopping at Blue is 38. P(B) = 38.

5

Substitute the probability of spinning a Red and the probability of spinning a Blue into the formula.

P(R or B) = 28 + 38

P(R or B) = 58

6

Simplify the fraction if possible. (The fraction in our example is already as simple as possible).

Answer:

The probability of spinning a Red or a Blue is 58.

We have expressed the probability as a fraction. We can also express this as a number (0.625) or a percentage (62.5%).

Slider

The slider below another real example of using the addition rule of probability.

In this example, the probability of three events is found using the addition rule.

Open the slider in a new tab

See Also

What is a fraction? Simplifying fractions What is a percentage?