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All 8 Flashcards — Bayes' theorem
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Question
What does Bayes' theorem do?
Answer
It reverses a conditional probability: turns P(evidence | cause), which you usually know, into P(cause | evidence), which you usually want.
Question
State the conditional-probability formula behind Bayes.
Answer
P(A | B) = P(A ∩ B) / P(B) — the joint probability of the wanted branch over the total probability of the condition.
Question
State Bayes' theorem (two-event form).
Answer
P(A | B) = [P(B|A)P(A)] / [P(B|A)P(A) + P(B|A′)P(A′)].
Question
What is the law of total probability for an event B?
Answer
P(B) = P(B|A)P(A) + P(B|A′)P(A′) — sum the probability of B over every branch.
Question
How do you build the denominator in a Bayes problem?
Answer
Add up every path on the tree that ends in the observed evidence (the total probability of the evidence).
Question
Why can P(disease | positive) be small even with a 95%-accurate test?
Answer
If the disease is rare, the many false positives from the large healthy group outnumber the few true positives, so a positive result is often a false alarm.
Question
What's the most common Bayes mistake?
Answer
Confusing P(A | B) with P(B | A) — the whole point of Bayes is that these two are different.
Question
How does Bayes change with three causes A₁, A₂, A₃?
Answer
The denominator becomes P(B|A₁)P(A₁) + P(B|A₂)P(A₂) + P(B|A₃)P(A₃); the method (wanted branch ÷ total) is unchanged.
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Topic 4.13 hub
Bayes' theorem (HL only)
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