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GIVEN:

𝑃(B) = 0.1

𝑃(A|B) = 0.8

𝑃(A|¬B) = 0.2

QUESTION: What is P(A)?

I know P(A∧B) = 𝑃(A|B)𝑃(B) = 0.08 but I can't figure out how to get P(A) from that and the given probabilities.

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3 Answers 3

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Hint:

Find $P(A\wedge B)$ exactly as you describe.

Find $P(A\wedge \neg B)$ in a similar fashion.

What is $P(A\wedge B)+P(A\wedge \neg B)$? (not just in terms of value, but in terms of what it represents)

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Use the law of total probability: $$P(A)=P(A|B)P(B)+P(A|\neg B)P(\neg B)$$ where $P(\neg B)=1-P(B)$.

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Note that $$ P(A) = P(A \cap B) + P(A \cap \lnot B) = P(A|B) P(B) + P(A|\lnot B) P(\lnot B) $$ and you know all the probabilities on the right hand side.

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