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Probability and conditional information

Probability and conditional information · Diagrams · Complete reading

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Complement, union and intersection

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Three source Venn constructions in universal set epsilon. Complement of A shades outside A. Union shades all of both overlapping circles A and B. Intersection shades only their overlap. Geometry and areas are schematic.
Three source Venn constructions in universal set epsilon. Complement of A shades outside A. Union shades all of both overlapping circles A and B. Intersection shades only their overlap. Geometry and areas are schematic.
Three source Venn constructions in universal set epsilon. Complement of A shades outside A. Union shades all of both overlapping circles A and B. Intersection shades only their overlap. Geometry and areas are schematic.
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Complement, disjointness and conditional reference sets

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Source proof sketches: E lies in sample space S, with its complement the rest; disjoint A and B have no common outcomes; for conditional probability given B, the entire B region is the reference set and its overlap with A is favourable. Blue marks the intersection and orange marks the rest of B. Areas are schematic, not numerical probabilities. The union double-counting overlap uses the preceding intersection construction.
Source proof sketches: E lies in sample space S, with its complement the rest; disjoint A and B have no common outcomes; for conditional probability given B, the entire B region is the reference set and its overlap with A is favourable. Blue marks the intersection and orange marks the rest of B. Areas are schematic, not numerical probabilities. The union double-counting overlap uses the preceding intersection construction.
Source proof sketches: E lies in sample space S, with its complement the rest; disjoint A and B have no common outcomes; for conditional probability given B, the entire B region is the reference set and its overlap with A is favourable. Blue marks the intersection and orange marks the rest of B. Areas are schematic, not numerical probabilities. The union double-counting overlap uses the preceding intersection construction.
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Read a two-stage probability tree

Explore a probability tree

Choose outcomes, change conditional probabilities, and compare path products with event totals.

Multiply along a complete path; add disjoint complete paths. Each reached split totals 1. Second-stage probabilities are conditional on the first event.

Start branches to A and not-A with probabilitiesP(A),P(not-A). From A branch to B or not-B with conditional probabilitiesP(B|A),P(not-B|A); from not-A branch to B or not-B withP(B|not-A),P(not-B|not-A). Four leaf equations multiply the first probability by its conditional second-stage probability to give each intersection probability. Sibling alternatives are mutually exclusive and exhaustive.
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