What it is (plain language)
The addition rule for probabilities gives a way to compute the chance that at least one of two events occurs. There are two forms: a simpler one when the two events cannot happen together (mutually exclusive), and a general one that applies whether or not they overlap.
Key definitions
– Event: a particular outcome or set of outcomes from an experiment (for example, “roll a 3” or “select a girl”).
– Probability: a number between 0 and 1 that measures how likely an event is to occur.
– Mutually exclusive events: two events that cannot both happen on the same trial (their overlap is empty).
– Intersection (A and B): the event that both A and B occur simultaneously.
– Union (A or B): the event that at least one of A or B occurs.
The formulas
– If A and B are mutually exclusive:
P(A or B) = P(A) + P(B)
(Because P(A and B) = 0 in this case.)
– In general (whether or not they are mutually exclusive):
P(A or B) = P(A) + P(B) − P(A and B)
Step-by-step checklist for using the rule
1. Clearly define events A and B and the sample space.
2. Compute or obtain P(A) and P(B).
3. Decide whether A and B are mutually exclusive:
– If yes, use P(A or B) = P(A) + P(B).
– If no, determine P(A and B) (the overlap).
4. Apply the general formula P(A or B) = P(A) + P(B) − P(A and B).
5. Verify the result lies between 0 and 1.
6. If probabilities are given as fractions, keep common denominators or convert to decimals for arithmetic.
Worked numeric examples
1) Mutually exclusive — single die roll
Event A = roll a 3, Event B = roll a 6. These cannot both occur on one roll, so they are mutually exclusive.
P(A) = 1/6, P(B) = 1/6.
P(A or B) = P(A) + P(B) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 0.3333.
2) General case with overlap — classroom example
Class composition: 9 boys, 11 girls (total 20). B-grade students: 4 boys and 5 girls (so 9 B students total).
Define A = “selected student is a girl”, B = “selected student has a B”.
P(A) = 11/20 = 0.55.
P(B) = 9/20 = 0.45.
P(A and B) = (girls with B) / total = 5/20 = 0.25.
Apply the general formula:
P(A or B) = P(A) + P(B) − P(A and B) = 0.55 + 0.45 − 0.25 = 0.75.
So there is a 75% chance a randomly chosen student is either a girl or a B student.
Notes and assumptions
– Probabilities must sum to values between 0 and 1.
– The “mutually exclusive” shortcut only applies when the intersection probability is exactly zero.
– Independence (one trial not affecting another) is a different concept: two independent events can still happen together; for independent events, P(A and B) = P(A)·P(B), which can be used to compute the intersection when appropriate.
Further reading (reputable sources)
– Investopedia — Addition Rule for Probabilities: https://www.investopedia.com/terms/a/additionruleforprobabilities.asp
– Khan Academy — Probability: https://www.khanacademy.org/math/statistics-probability/probability-library
– StatTrek — Addition Rule for Probabilities: https://stattrek.com/probability/addition-rule.aspx
– Wikipedia — Addition rule (probability): https://en.wikipedia.org/wiki/Addition_rule
– MIT OpenCourseWare — Intro to Probability and Statistics: https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2014/
Educational disclaimer
This explanation is for educational purposes only and is not personalized investment or legal advice.