Tree diagrams & conditional probability (every paper)

Multiply along, add across. Without replacement changes branch 2. Exam-style Factory: 60% machines A (2% faulty), 40% machines B (5% faulty) P(random item is faulty)? And P(it came from A | it is faulty)? P(faulty) = 0.6×0.02 + 0.4×0.05 = 0.012 + 0.020 = 0.032 P(A|faulty) = 0.012/0.032 = 0.375 ✓ 0.032 and 0.375 — multiply along paths, then restrict for the conditional. ℹ

"At least one" shortcut

P(at least one success in n tries) = 1 − P(no successes). For 3 free throws at 70%: 1 − 0.3³ = 0.973.

"At least one" vs "exactly one"

"At least one" = 1 − P(none) — the fast complement shortcut, covering one, two, or more successes.
"Exactly one" needs each individual path (success-fail-fail, fail-success-fail, fail-fail-success) added together — usually a longer calculation, not the complement shortcut.

Practice for this standard covers the whole topic — flashcards and quiz.