Independence: when events do not influence each other
Two events \(A\) and \(B\) are called independent if learning that one occurred does not change the probability of the other.
In everyday language, “independent” means the events have no probabilistic influence on each other.
The most common algebraic test for independence is
\[
P(A\cap B)=P(A)\,P(B).
\]
This identity is both necessary and sufficient: if it holds, \(A\) and \(B\) are independent; if it fails, they are dependent.
Connection to conditional probability
Independence can also be stated using conditional probability. If \(P(B)>0\), then
\[
P(A\mid B)=\frac{P(A\cap B)}{P(B)}.
\]
If \(A\) and \(B\) are independent, then \(P(A\mid B)=P(A)\). This is equivalent to the product rule:
multiplying both sides by \(P(B)\) gives \(P(A\cap B)=P(A)P(B)\).
The calculator focuses on the product test because it is direct when you already know \(P(A)\), \(P(B)\), and \(P(A\cap B)\).
What the checker is doing
The tool computes the product \(P(A)P(B)\) and compares it to the provided joint probability \(P(A\cap B)\).
Because real inputs are often rounded (for example, \(0.3333\) instead of \(1/3\)), the checker uses a small
tolerance and declares the values “equal” if the absolute difference satisfies \(|x-y|\le \text{tol}\).
If the equality holds within tolerance, the tool returns “independent (given inputs).”
If not, it returns “not independent (given inputs).”
Pairwise vs. mutual independence (3+ events)
With three or more events, there are multiple notions of independence. Events are pairwise independent if every pair
satisfies \(P(A_i\cap A_j)=P(A_i)P(A_j)\). Events are mutually independent if every joint probability factorizes
as a product, including higher-order intersections such as \(P(A\cap B\cap C)=P(A)P(B)P(C)\).
A classic university-level subtlety is that pairwise independence does not necessarily imply mutual independence.
That is why the multi-event mode can (1) test pairwise relations using provided pairwise joints, and (2) optionally test mutual independence if you also provide the full joint probability.
How to use this tool
In two-event mode, enter \(P(A)\), \(P(B)\), and \(P(A\cap B)\), then click Calculate. The calculator shows the product,
the difference, and a clear Yes/No verdict. In multi-event mode, list each marginal probability on its own line, then provide any
pairwise joint probabilities you have. The tool marks each pair as Pass/Fail/Needs info, and it can also check mutual independence if you provide the full joint.
The animation visualizes the equality as a product-vs-joint comparison bar and highlights the checklist items as the verification proceeds.
