Probability course
Probability course
Module 3 of 8

Independence

Separate independence, exclusion, and conditioning.

16 min2 worked examples3 concept checks

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Learning objectives
  • Test independence with an intersection probability.
  • Recognize that disjoint positive-probability events are dependent.
  • Distinguish pairwise independence from mutual independence.

Concepts

Independence is an equation

Events A and B are independent when P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B). If P(B)>0, this means learning B does not change the chance of A. Independence is a property of the probability model, not something established because two events sound unrelated.

Different from mutually exclusive

Mutually exclusive events cannot both happen: their intersection probability is zero. If each has positive probability, their product is positive, so they cannot be independent. Learning one occurred rules out the other. By contrast, independent events can occur together.

A group needs more than pairwise checks

For three events, mutual independence requires every pairwise factorization and the triple factorization. Pairwise independence alone can hide a constraint linking all three. Conditioning can also create dependence: independent dice become dependent once their sum is known.

Examples

Worked example 1

Parity on two dice

Roll two independent fair dice. Let A mean the first die is even and B mean the sum is odd. Are A and B independent?

  1. Find the marginal probabilities

    Half the first-die faces are even, so P(A)=1/2. The sum is odd when one die is even and the other odd, so P(B)=1/2.

  2. Count their intersection

    For A and B together, the first die is even and the second odd. There are 3×3=9 ordered pairs out of 36: P(A∩B)=1/4.

  3. Compare with the product

    1/4=(1/2)(1/2)1/4=(1/2)(1/2), so the events are independent. An event involving both dice can still be independent of an event on one die.
Worked example 2

Pairwise is not the whole story

Toss two independent fair coins. A is first coin heads, B is second coin heads, and C is both coins matching. Are the three events mutually independent?

  1. Check each pair

    Each event has probability 1/2. A∩B, A∩C, and B∩C each occur only on HH, with probability 1/4. Every pair is independent.

  2. Check the triple

    A∩B∩C is still HH, with probability 1/4. But P(A)P(B)P(C)=1/8.

  3. Identify the hidden relation

    The events are pairwise independent but not mutually independent. Once both individual coin outcomes are known, matching is determined.

Concept checks

Choose an answer, then check it. You can retry; your latest checked answer determines completion.

0 of 3 concept checks passed
Concept check 1Two mutually exclusive events each have probability strictly between 0 and 1. Can they be independent?
Concept check 2P(A)=0.4, P(B)=0.5, and P(A∩B)=0.2. Are A and B independent?
Concept check 3An urn has two red and two blue balls. Two draws are made without replacement. Does a red first draw change the probability of a red second draw?

Key results

  • Independence means probabilities factor.
  • Disjoint is usually the opposite of independent.
  • Check the full collection, not only each pair.
  • State whether conditioning changes the model.

Practice problems

Solve related problems or try a timed probability drill.

Practice problemsHints and worked solutions included.Timed probability drillShort questions on this module’s topic.
Next in the course

Choose a model that fits

Translate random experiments into useful distributions.

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