Question

Found all answers to 2 decimal places where appropriate, but keep as many as possible for intermediate computations. Suppose that

- P(A)=0.15
- P(B)=0.67,
- P(C)=0.38
- P(B∣A)=0.82,
- P(A∩C)=0, and that
- B and C are independent.
(a).Compute P(A∪C) :
(b).Compute P(B∩C) :
(c).Compute P(A∩B) :
(d).Compute P(A∪B) :

Answer 1

a. P(A∪C) = 0.53, b. P(B∩C) ≈ 0.2546, c. P(A∩B) ≈ 0.123, d. P(A∪B) ≈ 0.697

Step-by-step explanation:

a. To compute P(A∪C), we can apply the formula:

P(A∪C) = P(A) + P(C) - P(A∩C)

Given that P(A) = 0.15, P(C) = 0.38, and P(A∩C) = 0, we can substitute these values into the formula:

P(A∪C) = 0.15 + 0.38 - 0

P(A∪C) = 0.53

b. To compute P(B∩C), we can use the fact that B and C are independent. The formula for the intersection of two independent events is:

P(B∩C) = P(B) × P(C)

Given that P(B) = 0.67 and P(C) = 0.38, we can substitute these values into the formula:

P(B∩C) = 0.67 × 0.38

P(B∩C) ≈ 0.2546

c. To compute P(A∩B), we can use the formula for conditional probability:

P(A∩B) = P(A) × P(B∣A)

Given that P(A) = 0.15 and P(B∣A) = 0.82, we can substitute these values into the formula:

P(A∩B) = 0.15 × 0.82

P(A∩B) ≈ 0.123

d. To compute P(A∪B), we can apply the formula:

P(A∪B) = P(A) + P(B) - P(A∩B)

Given that P(A) = 0.15, P(B) = 0.67, and P(A∩B) ≈ 0.123 (computed in part c), we can substitute these values into the formula:

P(A∪B) = 0.15 + 0.67 - 0.123

P(A∪B) ≈ 0.697