Multi-event Probability: Addition Rule


The Addition Rule is used to calculate the probability that either (or both) of 2 events will happen:

Addition Rule Formula

When calculating the probability of either one of two events from occurring, it is as simple as adding the probability of each event and then subtracting the probability of both of the events occurring:

P(A or B) = P(A) + P(B) - P(A and B)
We must subtract P(A and B) to avoid double counting!

Special Case: Mutually Exclusive Events

If the 2 events are mutually exclusive (or sometimes called disjoint), that is if they can't BOTH happen!

If they both cannot happen, P(A and B) is 0. Therefore,

P(A or B) = P(A) + P(B)
Simplified formula when P(A and B) = 0, only useable when A and B are mutually exclusive events!

Intersection and Unions Visualized

We can illustrate the idea of mutually exclusive and non-mutually exclusive events using Venn Diagrams.

Example: Drawing a Queen of Hearts

Let's look at an example of how to use the Addition Rule to answer a question regarding a deck of 52 cards.

Q: What's the probability of drawing either a queen or a heart from deck of cards?

P(Q or ❤️) = P(Q) + P(❤️) - P(Q and ❤️)

In this case, you can get a card that’s both a queen and a heart (the queen of hearts!) so we need to subtract that probability to avoid double counting. There are 4 queens, 13 hearts, and one queen of hearts:

Mathematically:

  • P(Q or ❤️) = P(Q) + P(❤️) - P(Q and ❤️)

  • P(Q or ❤️) = 4/52 + 13/52 - 1/52

  • P(Q or ❤️) = 16/52 = 30.77%


Example Walk-Throughs with Worksheets

Video 1: Addition Rule Examples

Follow along with the workseet to work through the problem:

Practice Questions

Q1: The addition rule differs from the multiplication rule in that:
Q2: What is the probability of drawing a king or a spade from a deck of cards?
Q3: What is the probability of drawing a card from a deck and getting either a diamond or a spade?
Q4: What is the probability of rolling two dice and getting a sum of 5?
Q5: What is the probability of rolling a fair six sided die and getting either a 4 or 5?