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Test exact independence

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A sample space has 20 equally likely outcomes. Four outcomes are in both A and B. Find the intersection probability

\[\frac{4}{20}\]
Year 11, VCE Units 1 & 2, Chapter 8, 4 difficulty levels, Calculator allowedBuild a worksheet

Testing independence

Two events are independent when knowing one event does not change the probability of the other. Use an exact relation, such as the intersection probability equalling the product of the marginal probabilities

\[\Pr(A\cap B)=\Pr(A)\times\Pr(B)\]
  1. Find the intersection Calculate the probability that both events occur together \(\displaystyle \Pr(A\cap B)=\frac{4}{20}=\frac{1}{5}\)
  2. Find the marginal probabilities Calculate the probability of each event from the full sample space \(\displaystyle \Pr(A)=\frac{8}{20}=\frac{2}{5},\quad\Pr(B)=\frac{10}{20}=\frac{1}{2}\)
  3. Multiply the marginals The comparison value is the product of the separate event probabilities \(\displaystyle \Pr(A)\times\Pr(B)=\frac{2}{5}\times\frac{1}{2}=\frac{1}{5}\)
  4. Compare exact values Equal values mean independent and unequal values mean dependent \(\displaystyle \frac{1}{5}=\frac{1}{5}\Rightarrow\text{independent}\)

Worked example

A finite sample space has 24 equally likely outcomes. Event A contains 12 outcomes, event B contains 8 outcomes, and 4 outcomes are in both events. Test whether A and B are independent

\[\Pr(A\cap B)\mathrel{?=}\Pr(A)\times\Pr(B)\]
  1. \(\displaystyle \Pr(A\cap B)=\frac{4}{24}=\frac{1}{6}\)Use the 4 outcomes in both events over the 24 outcomes in the sample space
  2. \(\displaystyle \Pr(A)=\frac{12}{24}=\frac{1}{2}\)Use the 12 outcomes in event A over the full sample space
  3. \(\displaystyle \Pr(B)=\frac{8}{24}=\frac{1}{3}\)Use the 8 outcomes in event B over the full sample space
  4. \(\displaystyle \Pr(A)\times\Pr(B)=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}=\Pr(A\cap B)\Rightarrow\text{independent}\)The exact comparison values are equal, so the events are independent

Common mistakes

Using a sum test
✕\(\displaystyle \Pr(A\cap B)=\Pr(A)+\Pr(B)\)✓\(\displaystyle \Pr(A\cap B)=\Pr(A)\times\Pr(B)\)
The independence relation compares the intersection probability with the product of the marginal probabilities
Using the event descriptions
✕\(\displaystyle \text{Independent because the descriptions seem unrelated}\)✓\(\displaystyle \frac{4}{20}=\frac{8}{20}\times\frac{10}{20}\Rightarrow\text{independent}\)
Independence is a numerical relation, so it must be checked from exact probabilities rather than inferred from names or context
Calling exclusive events independent
✕\(\displaystyle \Pr(A\cap B)=0\Rightarrow\text{independent}\)✓\(\displaystyle \Pr(A\cap B)=0\ne\Pr(A)\times\Pr(B)\Rightarrow\text{dependent}\)
Two nonzero mutually exclusive events have a zero intersection but a positive product, so they are dependent

Difficulty levels

introIdentify independence in a small finite space using product equality
standardTest a count or probability table with an exact independence relation
challengeDistinguish independence from mutual exclusivity or test every pair in a three-event construction
examChoose a verdict supported by exact comparison values

Questions about this topic

What does independence mean?+

Two events are independent when the occurrence of one does not change the probability of the other

Why is the product used in the independence test?+

For independent events, the probability of both occurring equals the product of their separate probabilities

Are mutually exclusive events independent?+

Not when both events have positive probability. Their intersection is zero but the product of their marginal probabilities is positive

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 8: Probability and counting

In this chapter

Finite events, probability assignments and complementsRepresent a multi-stage finite sample spaceCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConditional probability from exact representationsDistribution of a finite random variableDesign and interpret a probability simulationCount restricted selections and calculate exact probabilitiesCount arrangements with product, sum and restriction rulesTest exact independenceExact probability of repeated independent successesMulti-stage probability, multiplication and total probabilityProbability from a uniform area modelProbabilities of sequences of independent events