Multiplication rule
Find a joint probability using a conditional probability.
Multiplication rule is one of 3 rules formulas in the probability section of this library, and it is used at high school · ap level.
Why multiplication rule works
For both events to happen, first B has to occur, and then A has to occur in the world where B already has. Chaining those two fractions gives the share of outcomes in the overlap. It is the conditional probability formula with the denominator moved across.
What each symbol means
$A\cap B$ means both events occur.
Multiplication rule: when it holds
For independent events this reduces to $P(A)P(B)$.
When it stops applying
The rule needs the conditioning event to have a positive probability, and it needs the conditional probability to be the genuine updated one. Sampling without replacement from a small group is the usual place this goes wrong, because the second draw is taken from a changed deck.
Multiplication rule: a worked example
If $P(B)=.4$ and $P(A\mid B)=.25$, then $P(A\cap B)=.10$.
The mistake to avoid
What people do: Multiplying the two plain probabilities when the second event depends on the first.
Why it goes wrong: Drawing two aces from a deck without replacement is 4 over 52 times 3 over 51, which is 1 in 221. Multiplying 4 over 52 twice gives 1 in 169, about 31 percent too likely.
Do this instead: Ask what has changed once the first event has happened. If the pool of possibilities shrank, use the updated conditional probability for the second factor.
Multiplication rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A\cap B$ means both events occur.
- Check the conditions before substituting. For independent events this reduces to $P(A)P(B)$.
- 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
- Rules
- 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 multiplication rule 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 multiplication rule
Does the order of the two events matter?
No. You can equally write it as the chance of B given A times the chance of A, and both routes give the same joint probability. Pick whichever conditional you actually know.
How do I extend this to three events?
Chain the conditions: the first event, then the second given the first, then the third given both. This is exactly what multiplying along a branch of a tree diagram does.
Does and always mean multiply?
Only when the second factor is properly conditioned. Multiplying unconditioned probabilities is a shortcut that is valid only for independent events.
When does this reduce to a simple product?
When the events are independent, so that the condition leaves the second probability unchanged. Two coin flips or two draws with replacement are the standard examples.