Exercise · ZW-BAYES-THEOREM-009 · ★★★ Advanced
If \(0<P(B)<1\) and \(P(A\mid B)=P(A\mid B^c)>0\), what follows for \(P(B\mid A)\)?
Choose one option.
Show answer
- D. \(P(B\mid A)=P(B)\)
Explanation
With common value \(p\), \(P(A)=p\) and Bayes gives \(\dfrac{p\,P(B)}{p}=P(B)\): learning \(A\) says nothing about \(B\).
Review the concept: Bayes' theorem
Bayes' theorem inverts a conditional probability: it gives the probability \(P(B_j\mid A)\) of a cause \(B_j\) given an observed effect \(A\), from the probability of the effect given that cause.
\[P(B_j\mid A)=\frac{P(A\mid B_j)\,P(B_j)}{\sum_{i}P(A\mid B_i)\,P(B_i)}\]
More practice problems: bayes' theorem
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