🎯 Conditional Probability and Independence (2D)

About this simulation

Every dot on the plane is one outcome of a random draw from a sample space Ω. Two translucent circles 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.

How it works

• P(A), P(B) — set each event's probability directly; the circle radii resize to match, since circle area ∝ probability.
• Dependence — slides the two circles apart or together. At 0 they sit at the distance that makes their overlap equal 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.
• Condition on B — restricts attention to the world where B already happened: points outside B fade and shrink away, and what remains shows P(A|B) = P(A∩B) / P(B) — the fraction of B that is also A.
• The panel compares the measured 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.

Controls

• P(A) / P(B) — probability of each event (5%–50%).
• Dependence — −1 (mutually exclusive) → 0 (independent) → +1 (one event nests inside the other).
• Condition on B — toggles the conditional view.
• ↺ Resample outcomes — redraws the random points.

Did you know?

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.