Testing event independence

Construct joint populations and distinguish independence from equal marginals or mutually exclusive events.

15 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Test independence of two events using their joint mass and compare a nonzero-evidence conditional probability with its marginal.

Prerequisites: Unequal outcome weights · Conditional draws · Complements and unions

Independence is a relation between events

Events A and B are independent when P(A and B)=P(A)P(B). If P(A)>0, this is equivalent to P(B given A)=P(B). Knowing A occurred then leaves the probability of B unchanged in the stated model. This is a mathematical relationship between specified events, not a claim that their names sound unrelated.

Independence does not require equal probabilities. One event could have probability 1/4 and another 1/2; independence would require intersection probability 1/8. Conversely two events can each have probability 1/2 and be strongly dependent. The marginals do not tell you which individual cases overlap.

Build the joint structure, not just two totals

Take eight equally likely labeled source cards. If two carry both A and B, two A only, two B only and two neither, each marginal is 4/8 and the intersection is 2/8. Multiplying the marginals gives 1/4, matching the intersection. Among the four A cases, two are B, so conditioning also gives 1/2.

Now move to a different population: four cards carry both labels and four neither. Both marginals remain 1/2, but the intersection is 1/2 rather than 1/4. Every A case is B, so P(B given A)=1. Equal marginal probabilities have not preserved independence. The activity keeps these supplied models separate; it does not reinterpret one observed sample as a proven distribution.

Keep the scope and zero boundary clear

Mutually exclusive events cannot occur together. If both have positive probability, their intersection is zero while their product is positive, so they are not independent. Do not use “independent” as a synonym for “disjoint.” The product definition also handles zero-probability events, whereas ordinary finite conditional division by a zero evidence mass is undefined.

A test for one pair of binary events is not proof that entire measurements, repeated samples or a collection of many events are mutually independent. Poker draws without replacement often create dependence through card removal. Check the actual event and information structure before multiplying probabilities as if independent. No finite set of observations establishes independence without a modeling or inferential argument.

Worked example

In 20 equally likely source cases, A occurs in 8, B in 5 and both in 2. Are these events independent?

P(A)=8/20=0.4 and P(B)=5/20=0.25.
P(A)P(B)=0.1.
P(A and B)=2/20=0.1, so the product criterion holds.
Since P(A)>0, check the conditional direction: P(B given A)=2/8=0.25=P(B).

These two events are independent in this supplied finite population. The conclusion does not establish independence of other events or of real repeated measurements.

Hands-on: Change the joint structure while preserving marginals

Construct both binary populations, select the whole event and evidence separately, then audit overlap, conditional and marginal fractions.

Reading population workshop…

Common mistakes

Using matching marginals as a proof of independence.
Replacing intersection mass with a product before checking independence.
Confusing disjoint events with independent events.
Dividing by zero evidence mass or generalizing a pairwise claim beyond its scope.

Keep these ideas

Independence means joint probability equals the product of marginals.
For nonzero evidence, compare conditional and marginal probability.
Equal marginal probabilities can coexist with dependence.
State which events and which supplied probability law the claim concerns.