Probability Rules — Venn Diagrams
P(A∪B), P(A|B) — none of it needs memorizing once you stop seeing formulas and start seeing two overlapping circles.
Probability Rules — Intuition with Venn Diagrams
Addition rule, multiplication rule, conditional probability — see them as areas before memorizing formulas.
P(A∪B) = P(A) + P(B) − P(A∩B) is just "area of two circles minus the overlap."
Conditional probability P(A|B) is "the fraction of B's circle occupied by A."
Press the Independence button to snap P(A∩B) = P(A)·P(B) — that's what independence means.
Draw one card from a 52-card deck. A = heart (13/52 = 0.25), B = face card (12/52 ≈ 0.23).
A∩B = heart face card (3/52 ≈ 0.06). → P(A∪B) = 0.25 + 0.23 − 0.06 = 0.42.
Independence example: two dice. A = 1st is even, B = 2nd is ≥3. The 1st roll doesn't affect the 2nd, so independent. P(A∩B) = 1/2 × 2/3 = 1/3.
- Step 1: Set P(A)=0.4, P(B)=0.3, P(A∩B)=0.12 → check the P(A∪B) readout below the graph: it shows 0.58. That's 0.4+0.3−0.12, the addition rule in action.
- Step 2: Press "Set Independent" → P(A∩B) auto-adjusts to P(A)·P(B). That's what independence means.
- Step 3: Drag P(A∩B) near 0 → the circles separate. This is "mutually exclusive" (can't happen together).
- Step 4: Push P(A∩B) close to P(B) → the P(A|B) readout approaches 1. If B happens, A almost certainly happens too.
▶ Interactive Venn Diagram
// Formula used here
Left: Conditional probability P(A|B)
• P(A∩B): probability both A and B happen (the overlap in a Venn diagram)
• ÷ P(B): restrict the universe to "B happened"
• Meaning: within the world where B occurred, what fraction also had A?
Right: Addition rule P(A∪B)
• P(A) + P(B): naively adding double-counts the overlap
• − P(A∩B): subtract it once to correct
• If mutually exclusive (A∩B = 0): just add — no overlap to worry about
Connection to independence
• A and B independent ⇔ P(A|B) = P(A) (knowing B doesn't change A's probability)
• Equivalent: P(A∩B) = P(A)·P(B) — the multiplication rule
// Common misconceptions
Mutually exclusive: A∩B = ∅ (rolling a 1 and a 2 on the same die).
Independent: P(A∩B) = P(A)P(B) (first roll being 1 doesn't affect the second roll).
Two mutually exclusive events (with P > 0) are never independent. If A happens, B's probability drops to 0 — that's the opposite of independence.
P(A|B) divides by P(B), not P(A∪B). You're restricting the universe to "B happened," so you divide by B's probability.
Only when A and B are mutually exclusive. In general, you must subtract P(A∩B) to avoid double-counting. Forgetting this can push probabilities above 1.
// Shapes you'll meet again
Around basic probability, the three Venn regions and a small set of recurring rewrites keep reappearing.
- The three-region Venn shape: once "A only," "B only," and "A∩B" are pinned down, P(A|B) and P(A∪B) become rearrangements of those three numbers
- The independence-check shape: with P(A)=0.3, P(B)=0.4, P(A∩B)=0.12, the product 0.3 × 0.4 lands at 0.12. "If multiplication recovers the joint, they're independent" is a check that appears on this single line
- The complement rewrite: "probability of at least one hit" reshapes into 1 − P(all misses). When cases pile up, flipping to the complement collapses them into one picture
- Mutually exclusive vs. independent as a contrast: "they don't co-occur" and "knowing one doesn't change the other" describe different structures. The very fact that they get confused is itself a recurring shape