Independence test
Check whether knowing one event changes the probability of another.
Independence test is one of 2 events formulas in the probability section of this library, and it is used at high school · ap level.
Why independence test works
Independence means learning that B happened changes nothing about A, so the conditional probability equals the plain one. Multiplying both sides of that statement by the probability of B converts it into a product of two plain probabilities, which is symmetric in the two events and still makes sense when a probability is zero.
What each symbol means
$A,B$ are events.
Independence test: when it holds
Do not confuse independence with mutually exclusive events; nonempty disjoint events are generally dependent.
When it stops applying
Checking every pair is not enough for three or more events. Take the first flip landing heads, the second landing heads, and the two flips matching: each pair is independent with joint probability 0.25, but all three together have joint probability 0.25 rather than the 0.125 that full independence requires.
Independence test: a worked example
If $P(A)=.5,P(B)=.4,P(A\cap B)=.2$, then $A$ and $B$ are independent.
The mistake to avoid
What people do: Calling two mutually exclusive events independent.
Why it goes wrong: If two events cannot both happen, their joint probability is 0 while the product of two positive probabilities is not. Disjoint events with nonzero probability are always dependent, because learning one occurred rules the other out completely.
Do this instead: Test the product rather than reasoning from words. Compute both sides and compare them; equality is the definition and nothing else is.
Independence test: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A,B$ are events.
- Check the conditions before substituting. Do not confuse independence with mutually exclusive events; nonempty disjoint events are generally dependent.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Events
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where independence test comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about independence test
Are independent and mutually exclusive the same thing?
No, they are close to opposites. Mutually exclusive means one event tells you the other did not happen, which is the strongest possible dependence, while independence means one tells you nothing.
How do I check independence from a two-way table?
Compare the conditional percentages across rows. If every row shows the same breakdown as the overall column totals, the two variables are independent in that table.
Does independence work in both directions?
Yes, the product form is symmetric, so if A carries no information about B then B carries none about A. That symmetry is one reason this version is preferred to the conditional one.
Can events be independent in pairs but not as a group?
Yes, as the two coin flips and their match show. Full independence requires the product rule to hold for every subset, not merely for every pair.