Every dot on the floor is one outcome of a random draw from a sample space Ω.
Two translucent discs mark event A (blue) and event B
(amber); a dot's colour shows whether it lands in A only, B only, both, or neither.
Because outcomes are scattered uniformly, the fraction of dots inside a region is a
live, countable estimate of that event's probability — P(A) ≈ points in A / total.
P(A)·P(B) — the signature of independence. Negative values push them apart toward mutual exclusivity; positive values pull B inside A (or A inside B) toward full dependence.P(A|B) = P(A∩B) / P(B) — the fraction of B that is also A.P(A∩B) against P(A)×P(B) live: when they match, A and B are independent — knowing B happened tells you nothing new about A.
Independence is a much stronger claim than "unrelated-sounding". Two events can be
statistically independent while being causally connected, and two causally linked
events can still satisfy P(A∩B)=P(A)P(B) by coincidence for one particular
pair of probabilities — independence is a numerical property of a specific
distribution, not a guarantee about mechanism.
A 3D scatter of sample-space outcomes sits beneath two overlapping event discs, A and B. Reshaping their size and overlap — and zooming into B — turns conditional probability and independence from formulas into something you can literally count and watch shrink.
Each dot is an outcome; a region's measured probability is just the fraction of dots inside it. P(A∩B) versus P(A)×P(B) tells you whether A and B are independent, and conditioning on B — shrinking every point outside it — makes P(A|B) visible as the share of B that's also A.
Set P(A) and P(B) to resize the two discs, then drag Dependence from −1 (mutually exclusive) through 0 (independent) to +1 (nested). Toggle "Condition on B" to zoom the picture down to the world where B has already happened.
Independence is a numerical coincidence of a particular pair of probabilities, not a promise about cause and effect — two causally linked events can still satisfy P(A∩B)=P(A)P(B) for one specific setting, then stop being independent the moment either probability changes.