Addition rule

College/University

Definition

Also known as the "sum rule." It is a rule for calculating the probability of two events A and B, that can be mutually exclusive or non-mutually exclusive. The formula P(A or B) = P(A) + P(B) is used for mutually exclusive events, and P(A or B) = P(A) + P(B) – P(A and B) for events that are non-mutually exclusive. For example, to calculate the probability of rolling a 1 or 2 on a 6 sided die we would simply add up the probability of each event as 1/6 + 1/6 = 1/3. This event is mutually exclusive because it is impossible to roll both numbers at the same time with one die. Example 2, calculating the probability of drawing a queen or daimond card from a 52 card deck. This event is non-mutually exclusive because there is a possibility of drawing a queen of diamonds which satisfies both events. So the second formula must be used to subtract the queen of diamonds from being counted twice.

Worked examples

\(P(\)1 or 2\() = P(\)1\() + P(\)2\() = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}\)

Rolling a 1 or 2 on one die — mutually exclusive events, so just add their probabilities.

\(P(\)Queen or Diamond\() = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}\)

Drawing a queen or diamond — non-mutually exclusive, so subtract the overlap (queen of diamonds).

Common mistakes

\(P(\)Queen or Diamond\() = \frac{4}{52} + \frac{13}{52} = \frac{17}{52}\) → \(P(\)Queen or Diamond\() = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52}\) Forgot to subtract the queen of diamonds, which is counted in both groups.
\(P(\)1 or 2\() = \frac{1}{6} + \frac{1}{6} - P(\)1 and 2\()\) → \(P(\)1 or 2\() = \frac{1}{6} + \frac{1}{6}\) These are mutually exclusive — you cannot roll both at once, so no overlap to subtract.
Using \(P(A) + P(B) - P(A \) and \( B)\) for every problem → Use \(P(A) + P(B)\) when events are mutually exclusive The subtraction is only needed when overlap is possible; check if events can happen together.

Where you'll use it next

The addition rule is foundational for compound probability, Venn diagrams, and conditional probability in statistics, and prepares you for the multiplication rule and Bayes' theorem in advanced probability courses.

Found in 1 StudyPug lesson

Understanding the Addition Rule for 'OR' in Probability

Statistics

Discover how to calculate probabilities for 'OR' scenarios using the addition rule. Learn to distinguish between mutually exclusive and non-mutually exclusive events for accurate probability assessments.

12th Grade12th