The accompanying tree diagram represents a two-stage experiment. (a) $P(A \cap D)$ (b) $P(B \cap D)$ (c) $P(C \cap D)$ (d) $P(D)$ (e) Verify. $P(A \cap D)$ $P(A | D) = \frac{P(A \cap D)}{P(D)} = \frac{P(A) \cdot P(D | A)}{P(A) \cdot P(D | A) + P(B) \cdot P(D | B) + P(C) \cdot P(D | C)}$
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$P(A) = 1/9$ $P(B) = 4/9$ $P(C) = 4/9$ $P(D|A) = 1/2$ $P(D|B) = 1/3$ $P(D|C) = 3/4$ $P(A \cap D) = P(A) \times P(D|A) = (1/9) \times (1/2) = 1/18$ $P(B \cap D) = P(B) \times P(D|B) = (4/9) \times (1/3) = 4/27$ $P(C \cap D) = P(C) \times P(D|C) = (4/9) \times (3/4) Show more…
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