Exercise · ZW-LAW-OF-TOTAL-PROBABILITY-001 · ★☆☆ Basic
Events \(B_1,B_2\) form a partition with \(P(B_1)=0.4\), \(P(B_2)=0.6\), \(P(A\mid B_1)=0.5\) and \(P(A\mid B_2)=0.2\). What is \(P(A)\)?
Choose one option.
Show answer
- B. 0.32
Explanation
Weight each conditional probability: \(0.4\cdot0.5+0.6\cdot0.2=0.2+0.12=0.32\). Averaging 0.5 and 0.2 (0.35) ignores the weights.
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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