The multiplication rule for probabilities
When you want the probability that multiple events all happen, you are asking for an intersection such as
\(P(A\cap B)\) (“A and B”). The multiplication rule connects intersections to conditional probabilities:
\[
P(A\cap B)=P(A)\,P(B\mid A).
\]
This statement is always true as long as \(P(A)>0\). It comes directly from the definition of conditional probability,
\(P(B\mid A)=\dfrac{P(A\cap B)}{P(A)}\), rearranged to solve for \(P(A\cap B)\).
Independent vs. dependent events
Two events \(A\) and \(B\) are independent when knowing \(A\) occurred does not change the probability of \(B\).
Formally, independence can be written as \(P(B\mid A)=P(B)\) (and also \(P(A\mid B)=P(A)\) when probabilities are positive).
Substituting \(P(B\mid A)=P(B)\) into the multiplication rule gives the familiar independent form:
\[
P(A\cap B)=P(A)\,P(B).
\]
If the events are dependent, then you must use the conditional probability \(P(B\mid A)\), which may be larger or smaller than \(P(B)\).
A classic example is drawing cards without replacement: the probability of the second draw depends on what happened on the first draw.
Chains of events (multi-stage processes)
The same idea extends to more than two events. For three events,
\[
P(A_1\cap A_2\cap A_3)=P(A_1)\,P(A_2\mid A_1)\,P(A_3\mid A_1\cap A_2).
\]
In general, for a chain of \(n\) events,
\[
P(A_1\cap A_2\cap \cdots \cap A_n)=P(A_1)\prod_{i=2}^{n} P\!\left(A_i \mid A_1\cap \cdots \cap A_{i-1}\right).
\]
In the independent special case, every conditional term equals the corresponding marginal probability, so the chain collapses to a simple product:
\[
P(A_1\cap \cdots \cap A_n)=\prod_{i=1}^{n} P(A_i).
\]
This is why “multi-stage” problems often reduce to multiplying step probabilities—each link in the chain contributes one factor between 0 and 1.
How to use this tool
Choose whether your situation is dependent or independent. Then select how many events are in your chain.
In independent mode, enter the marginals \(P(A_i)\). In dependent mode, enter \(P(A_1)\) and the conditional probabilities for later steps,
such as \(P(A_2\mid A_1)\) and \(P(A_3\mid A_1\cap A_2)\). Click Calculate to see the product computed step by step,
along with a decimal and percent result. The interactive diagram visualizes the chain as a sequence of links; pressing Play
animates how probability “flows” through the links to form the final intersection probability.
Advanced note (university level)
In stochastic processes, this chaining idea appears in joint distributions and in probability trees. For example, Markov chains use the special structure
\(P(X_{t+1}\mid X_t)\), which simplifies long products of conditionals. The multiplication rule is also fundamental in measure-theoretic probability,
where conditional probability can be formalized using conditional expectations.
