Addition Rule of Probability
In short: The addition rule finds the chance of A or B by adding P(A) and P(B) and then removing the shared part, P(A and B), so outcomes lying in both events are not tallied twice.
Algebra 2ProbabilityS-CP.B.7
What is Addition Rule of Probability?
The addition rule measures how likely it is that one event, the other, or both will occur. Adding the two probabilities alone double-counts whatever sits in both events, so one copy of that shared piece has to come back out.
The formula is \(P(A \text{ or } B) = P(A) + P(B) – P(A \text{ and } B)\). When the events are mutually exclusive the overlap is zero, and the formula shrinks to \(P(A) + P(B)\). Here “or” is inclusive, meaning A, B, or both.
How to spot it
- The word or joins two events, and some outcomes can satisfy both descriptions at once.
- A two-way table or Venn diagram shows a count in the overlap, such as students in both band and choir.
- A deck-of-cards question asks for hearts or face cards, where the jack, queen and king of hearts count in each.
- You are given P(A), P(B) and P(A or B) and asked for the missing overlap.
The method
- Find \(P(A)\), \(P(B)\) and the overlap \(P(A \text{ and } B)\) from the counts or the diagram.
- Add \(P(A)\) and \(P(B)\).
- Subtract the overlap once.
- Simplify, and check that the result is between 0 and 1 and at least as large as each single probability.
Addition Rule of Probability: solved example
Problem
A school with 120 students has 48 in band, 35 in choir and 14 in both groups. A name is drawn from a hat. How likely is it to belong to someone in band, choir, or both?
Solution
Use \(P(\text{band}) = \frac{48}{120}\), \(P(\text{choir}) = \frac{35}{120}\) and \(P(\text{both}) = \frac{14}{120}\).
Then \(P(\text{band or choir}) = \frac{48 + 35 – 14}{120} = \frac{69}{120} = \)\(\frac{23}{40}\).
Try it
Question 1
A card is drawn from a standard 52-card deck. Find \(P(\text{heart or face card})\).
Show the answer
\(\frac{13}{52} + \frac{12}{52} – \frac{3}{52} = \frac{22}{52} = \frac{11}{26}\).
Question 2
Given \(P(A) = 0.5\), \(P(B) = 0.3\) and \(P(A \text{ or } B) = 0.65\), find \(P(A \text{ and } B)\).
Show the answer
\(0.65 = 0.5 + 0.3 – x\), so \(x = 0.15\).
Addition Rule of Probability vs. Mutually Exclusive Events
The addition rule is the general formula, and it has a subtraction for overlap. The exclusive case is a special situation where the overlap is zero, so only the simple sum remains. Using the short version on events that do overlap overstates the probability, sometimes beyond 1.
Keep going
Lessons
Worksheets and quizzes
- Conditional Probability and Independence – Worksheet
- Conditional Probability and Independence – Quiz
- Probability Rules and Simulation – Worksheet
- Probability Rules and Simulation – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Probability