Answer:
The answer is below
Step-by-step explanation:
P(A') = 1 - P(A) = 1 - 0.3 = 0.7
P(B'/A) = 1 - P(B/A) = 1 - 0.75 - 0.25
P(B/A') = 0.2, P(B'/A') = 1 - 0.2 = 0/8
P(C|A∩B) = 0.20,
P(C|A'∩B) = 0.15,
P(C|A∩B') = 0.80, and
P(C|A'∩B') = 0.90.
A) From conditional probability;
P(B/A) = P(B ∩ A) / P(A)
P(B∩A) = P(B/A) × P(A) = 0.75 × 0.3 = 0.225
P(C/A∩B) = P(A∩B∩C)/P(A∩B)
P(A∩B∩C) = P(C/A∩B) × P(A∩B)
P(A∩B∩C) = 0.2 ×0.225 = 0.045
B) P(A∩B∩C) = P(A) × P(B/A) × P(C/A∩B)
P(B'∩C) = P(A∩(B'∩C)) + P((A'∩B')∩C)
P(B'∩C) = P(A) × P(B'/A) × P(C/A∩B') + P(A') × P(B'/A') × P(C/A'∩B')
P(B'∩C) = 0.3 × 0.25 × 0.8 + (0.7 × 0.8 × 0.9) = 0.06 + 0.504 = 0.564
C) P(C) = P(A∩B∩C) + P(A'∩B∩C) + P(A∩B'∩C) + P(A'∩B'∩C)
P(A'∩B∩C) = P(A') × P(B/A') × P(C/A'∩B) = 0.7 × 0.2 × 0.15 = 0.021
P(A∩B'∩C) = P(A) × P(B'/A) × P(C/A∩B') = 0.3 × 0.25 × 0.8 = 0.06
P(A'∩B'∩C) = P(A') × P(B'/A') × P(C/A'∩B') = 0.7 × 0.8 × 0.9 = 0.504
P(C) = P(A∩B∩C) + P(A'∩B∩C) + P(A∩B'∩C) + P(A'∩B'∩C) = 0.045 + 0.021 + 0.06 + 0.504 = 0.63
D) P(A/B'∩C) = P(A∩B'∩C)/P(B'∩C)
P(A/B'∩C) = 0.06 / 0.564 = 0.106
