The General Addition Law
The General Addition Law is the rule P(A or B) = P(A) + P(B) - P(A and B), which gives the probability that at least one of two events happens by adding the two individual probabilities and then subtracting the probability that both happen, so the overlap is not counted twice. It is used in GCSE Statistics Higher tier questions involving Venn diagrams and two-way tables.
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Method
- Identify the two events A and B and write down each individual probability, P(A) and P(B), from the table, Venn diagram or description given.
- Work out P(A and B), the probability that both events happen at the same time. This is the overlap region on a Venn diagram or the combined cell in a two-way table.
- Substitute the values into the formula P(A or B) = P(A) + P(B) - P(A and B).
- If the question states or implies the events are mutually exclusive, use P(A and B) = 0, so the formula simplifies to P(A or B) = P(A) + P(B).
- Simplify the resulting fraction or decimal fully.
- Check the answer is sensible: it must lie between 0 and 1, and P(A or B) cannot be smaller than either P(A) or P(B) on its own.
Worked example
In a class of 30 students, 18 study French, 14 study Spanish, and 7 study both French and Spanish. A student is picked at random. Find the probability that the student studies French or Spanish.
- P(French) = 18/30
- P(Spanish) = 14/30
- P(French and Spanish) = 7/30, since 7 students study both
- Apply the General Addition Law: P(French or Spanish) = P(French) + P(Spanish) - P(French and Spanish) = 18/30 + 14/30 - 7/30 = 25/30
- Simplify 25/30 by dividing top and bottom by 5
- Final answer: P(French or Spanish) = 5/6
Practice questions
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Q1P(A) = 0.4, P(B) = 0.3, P(A and B) = 0.1. Find P(A or B).Show answer
Answer: 0.6 (0.4 + 0.3 - 0.1)
Q2P(A) = 1/2, P(B) = 1/3, P(A and B) = 1/6. Find P(A or B).Show answer
Answer: 2/3 (3/6 + 2/6 - 1/6 = 4/6 = 2/3)
Q3Out of 50 people surveyed, 20 like tea, 25 like coffee, and 10 like both. Find the probability that a randomly chosen person likes tea or coffee.Show answer
Answer: 7/10 (20/50 + 25/50 - 10/50 = 35/50 = 7/10)
Q4P(A) = 0.55 and P(B) = 0.4. Events A and B are mutually exclusive. Find P(A or B).Show answer
Answer: 0.95 (mutually exclusive, so P(A and B) = 0, giving 0.55 + 0.4)
Q5In a group of 40 students, 22 play football, 18 play rugby, and P(football or rugby) = 0.85. Find the number of students who play both sports.Show answer
Answer: 6 students (0.85 = 0.55 + 0.45 - P(both), so P(both) = 0.15, and 0.15 x 40 = 6)
Q6P(A) = 0.6, P(A or B) = 0.75, and P(A and B) = 0.2. Find P(B).Show answer
Answer: 0.35 (0.75 = 0.6 + P(B) - 0.2, so P(B) = 0.75 - 0.4)
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
In a school of 200 students, 90 study Art, 70 study Music, and 25 study both Art and Music. A student is chosen at random. Work out the probability that the student studies Art or Music.
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Events A and B are such that P(A) = 0.45, P(B) = 0.35 and P(A or B) = 0.68. (a) Find P(A and B). (b) State, with a reason, whether A and B are mutually exclusive.
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A vet surveys 120 pet owners. 64 own a dog, 52 own a cat, and the probability that a randomly chosen owner owns neither a dog nor a cat is 0.2. (a) Find the number of owners who own both a dog and a cat. (b) Find the probability that a randomly selected owner owns a dog or a cat but not both.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the the general addition law worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 21 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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