Multiplication rule for dependent events
When two events are dependent, the probability that they both occur is not usually the product of their separate probabilities.
Dependence means that learning one event happened can change the probability of the other. The correct general relationship is the
multiplication rule:
\[
P(A\cap B)=P(A)\,P(B\mid A).
\]
The term \(P(B\mid A)\) is the conditional probability of \(B\) given that \(A\) occurred, and it captures the dependence between the events.
If \(A\) makes \(B\) more likely, then \(P(B\mid A) > P(B)\); if it makes \(B\) less likely, then \(P(B\mid A) < P(B)\).
Why the formula is true
One way to understand the rule is to think in “stages.” First, the experiment must land in event \(A\).
That happens with probability \(P(A)\). Once you are inside \(A\), only some outcomes also belong to \(B\).
The fraction of outcomes in \(A\) that are also in \(B\) is \(P(B\mid A)\). Multiplying these stages gives the probability of landing in both:
\[
\text{(probability of being in \(A\))}\times\text{(probability of also being in \(B\) once in \(A\))}.
\]
This also connects to the definition \(P(B\mid A)=\frac{P(A\cap B)}{P(A)}\) (when \(P(A)>0\)), which rearranges directly into
\(P(A\cap B)=P(A)P(B\mid A)\).
Independent events as a special case
Independence is a special situation where knowing \(A\) happened does not change the probability of \(B\).
In that case \(P(B\mid A)=P(B)\), so the multiplication rule becomes the familiar formula
\[
P(A\cap B)=P(A)P(B).
\]
The calculator allows you to enter either \(P(B\mid A)\) (general/dependent) or \(P(B)\) (independent shortcut).
If you select the independent option, the tool interprets your input as \(P(B)\) and uses \(P(B\mid A)=P(B)\).
Chains of events (more than two)
The multiplication idea extends naturally to several events. For a sequence \(A_1, A_2, \dots, A_n\),
the joint probability is
\[
P(A_1\cap A_2\cap \cdots \cap A_n)
= P(A_1)\,P(A_2\mid A_1)\,P(A_3\mid A_1\cap A_2)\cdots P(A_n\mid A_1\cap\cdots\cap A_{n-1}).
\]
This is called the chain rule for probability. It is especially useful in real problems where events occur step-by-step,
such as drawing cards without replacement, quality-control sampling, or successive system failures in engineering reliability.
How to use this tool
In two-event mode, enter \(P(A)\) and either \(P(B\mid A)\) or \(P(B)\) (if independent). The tool computes \(P(A\cap B)\),
shows a clear step-by-step multiplication, and can display percent form. In chain mode, enter one probability per line
(optionally labeling conditional information with “\(|\)”) and the tool multiplies the factors in order.
The interactive diagram visualizes the product as a sequence of “gates” that refine the running joint probability.
Press Play to animate the chain and see how each conditional term affects the final result.
