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Trees
Branches: Each node's outgoing probabilities sum 1
Paths: Multiply along path; add separate exclusive paths
At least one: 1−P(no successes) can be quicker
Draws
Independent: One event leaves other probability unchanged
Replacement: Return restores bag; Higher: no replacement changes counts
Both orders: With replacement: independent; Higher: add RB and BR
Two-way tables
Totals: Fill missing counts; rows, columns and grand total agree
Sample space: Whole group denominator vs specified subgroup
Conditional
Given B: Higher: P(A|B)=P(A∩B)/P(B), P(B)>0; restrict to B
Tree / counts: Higher: Later branch already conditional; flagged defectives 90/135
Direction: Higher: P(defective|flagged) differs from P(flagged|defective)
Connections
Trees → Draws: A path uses probabilities conditional on its route
Two-way tables → Conditional: A specified subgroup changes the sample space
