Key takeaway: The addition rule of probability gives the chance that event A or event B happens by adding both chances and subtracting the overlap, so shared outcomes count once.
Grade 11 – Algebra 2S-CP.B.7Probability and Counting
Definition of Addition Rule of Probability
The addition rule finds the probability that at least one of two events happens. Adding the two separate probabilities counts every outcome in the overlap twice, once for each event, so the overlap must be subtracted once.
Think of two clubs with some shared members. To count everyone belonging to either club, you add the club sizes and then remove the members you listed twice. Probabilities work the same way, with fractions of the sample space in place of headcounts. If the events cannot overlap at all, the subtracted term is zero and the rule reduces to plain addition.
Formula and notation
For any two events \(A\) and \(B\), with union \(\cup\) meaning “or” and intersection \(\cap\) meaning “and”:
\[P(A\cup B)=P(A)+P(B)-P(A\cap B).\]
“Or” here is inclusive: it includes the case where both happen.
Key facts
- The word “or” in probability includes the case where both events occur.
- A Venn diagram with counts in each region makes the overlap easy to see.
- The result can never exceed 1; an answer above 1 signals a forgotten subtraction.
- For disjoint events the overlap is 0, giving the simpler sum P(A) + P(B).
Addition Rule of Probability example
At a bike-safety check, 80 riders are inspected: 46 wear helmets, 38 have front lights and 21 have both. One rider is chosen at random. What is the probability the rider has a helmet or a light?
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Write the separate probabilities: \(P(\text{helmet})=\frac{46}{80}\), \(P(\text{light})=\frac{38}{80}\), \(P(\text{both})=\frac{21}{80}\).
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Apply the rule: \(\frac{46}{80}+\frac{38}{80}-\frac{21}{80}=\frac{63}{80}\).
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Convert: \(\frac{63}{80}=0.7875\).
The probability is \(\frac{63}{80}=0.7875\), about 78.8%.
Common mistake
Watch out
Kofi adds \(\frac{46}{80}+\frac{38}{80}=\frac{84}{80}\) and writes it as the answer.
Why it is wrong
A probability above 1 is impossible. The 21 riders with both items were counted in each group, so the addition rule subtracts them once: \(\frac{84-21}{80}=\frac{63}{80}\).
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Probability and Counting