Key takeaway: Independent events are two events where knowing that one happened does not change the probability of the other, so the probability that both occur is the product of their separate probabilities.
Grade 10 – GeometryS-CP.A.2Probability and Counting
Definition of Independent Events
Two events are independent events when the result of one has no effect on the chance of the other. Flipping a coin and then rolling a die is a classic pair: the coin cannot influence the die.
The test is a multiplication check. If the chance of both equals the product of the two separate chances, they are independent. If not, the events are dependent, and the first outcome shifts the second chance, as when you draw two cards without putting the first one back. Independence is about probabilities, not about whether events can happen together.
Formula and notation
Events \(A\) and \(B\) are independent exactly when:
\[ P(A \text{ and } B) = P(A)\cdot P(B) \]
Equivalent statement: \(P(A \mid B) = P(A)\). Knowing \(B\) leaves the probability of \(A\) unchanged.
Key facts
- Multiply the separate probabilities to get the chance that both happen.
- Drawing without replacement usually makes events dependent.
- Independent events can still occur together; they are not mutually exclusive.
- In a tree diagram, the second-stage branches repeat the same probabilities.
Independent Events example
At a school, 40% of students play soccer, 35% are in the band, and 14% do both. Are playing soccer and being in the band independent events? Then test a second school with 40%, 30% and 15% for the same three quantities.
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First school: \(P(\text{soccer})\cdot P(\text{band}) = 0.40 \times 0.35 = 0.14\).
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That equals the given overlap of 0.14, so the product test passes.
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Second school: \(0.40 \times 0.30 = 0.12\), but the overlap is 0.15. The product test fails.
At the first school the events are independent; at the second school they are dependent.
Common mistake
Watch out
Jun says two events that cannot happen together, like rolling a 2 and rolling a 5 on one die, are independent because they do not affect each other.
Why it is wrong
Events that cannot both happen are mutually exclusive, which is a different idea. Once you know the die shows a 2, the chance of a 5 drops to 0, so one outcome changes the other. Check: \(P(2 \text{ and } 5) = 0\), not \(\frac{1}{6}\cdot\frac{1}{6}\).
Related terms and lessons
- Conditional Probability
- Probability
- Compound Event
- Tree Diagram
- Mutually Exclusive Events
- Sample Space
- Geometry · Probability: Independence, Conditional Probability and Fair DecisionsLessonWorksheetQuiz
- Algebra 2 · Probability: Sample Spaces, Independence and Conditional ProbabilityLessonWorksheetQuiz
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