Exercise · ZW-LAW-OF-TOTAL-PROBABILITY-009 · ★★★ Advanced
If \(0<P(B)<1\) and \(P(A\mid B)=P(A\mid B^c)=p\), what is \(P(A)\)?
Choose one option.
Show answer
- B. \(p\)
Explanation
\(P(A)=pP(B)+p(1-P(B))=p\): if the outcome does not depend on the case, the case does not matter.
Review the concept: Law of total probability
The total probability of an event \(A\) is the weighted average of its conditional probabilities \(P(A\mid B_i)\) over a partition \(B_i\) of the sample space, with the probability of each case as its weight.
\[P(A)=\sum_{i=1}^{n}P(A\mid B_i)\,P(B_i)\]
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