Independence
Objective. Test and use independence through multiplication laws.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Two events are independent if the occurrence of one does not affect the probability of the other, formally tested by checking whether P(A∩B) = P(A)×P(B); if this equation holds, the events are independent, and equivalently P(A|B) = P(A). This multiplication law provides a quick way to calculate the probability of several independent events all occurring by multiplying their individual probabilities together. For example, if P(A) = 0.3 and P(B) = 0.5 are independent, then P(A∩B) = 0.3×0.5 = 0.15, and this value can be checked against given data to confirm independence. Independence must be tested numerically, never simply assumed from the context of a problem.
Worked example
Given P(A) = 0.4, P(B) = 0.25 and P(A∩B) = 0.1, determine whether events A and B satisfy independence.
- 1Recall the independence test: P(A∩B) should equal P(A)×P(B).
- 2Compute P(A)×P(B) = 0.4×0.25 = 0.1.
- 3Compare with the given P(A∩B) = 0.1.
- 4Since both values match, A and B are independent.
Answer: Yes, A and B are independent since P(A∩B) = P(A)×P(B) = 0.1
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Practice
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Start practisingCommon mistake. Students often assume events are independent just because they seem unrelated in context, without checking P(A∩B) = P(A)×P(B) numerically. Independence must always be verified by calculation, not assumed from a story description alone.
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