The accompanying tree diagram represents a two-stage experiment. (Let x = 0.45, y = 0.55, r = 0.65, s = 0.35, t = 0.4, and w = 0.6.)

Use the diagram to find the following probabilities. (Round your answers to three decimal places.)
(a) Find P(A) · P(D | A).


(b) Find P(B) · P(D | B).


(c) Find P(A | D).

(a) P(A) · P(D | A) = 0.450 · 0.400 = 0.180

(b) P(B) · P(D | B) = 0.550 · 0.600 = 0.330
(c) P(A | D) = P(A) · P(D | A) / [P(A) · P(D | A) + P(B) · P(D | B)] = 0.180 / (0.180 + 0.330) = 0.353

To find the probabilities using the tree diagram, we can follow these steps:

(a) P(A) · P(D | A):

1. Start at the top of the tree diagram. The first stage has two branches: A and B, with probabilities P(A) = 0.45 and P(B) = 0.55, respectively.
2. Follow the branch labeled A to the second stage. In the second stage, there are two branches: D and C, with probabilities P(D | A) = 0.65 and P(C | A) = 0.35, respectively.
3. Multiply the probabilities from the first and second stages together: P(A) · P(D | A) = 0.45 * 0.65 = 0.292.

Therefore, P(A) · P(D | A) is 0.292.

(b) P(B) · P(D | B):

1. Start at the top of the tree diagram. The first stage has two branches: A and B, with probabilities P(A) = 0.45 and P(B) = 0.55, respectively.
2. Follow the branch labeled B to the second stage. In the second stage, there are two branches: D and E, with probabilities P(D | B) = 0.4 and P(E | B) = 0.6, respectively.
3. Multiply the probabilities from the first and second stages together: P(B) · P(D | B) = 0.55 * 0.4 = 0.22.

Therefore, P(B) · P(D | B) is 0.22.

(c) P(A | D):

1. Start at the top of the tree diagram. The first stage has two branches: A and B, with probabilities P(A) = 0.45 and P(B) = 0.55, respectively.
2. Follow the branch labeled A to the second stage. In the second stage, there are two branches: D and C, with probabilities P(D | A) = 0.65 and P(C | A) = 0.35, respectively.
3. To find P(A | D), we use Bayes' theorem:
P(A | D) = P(A) * P(D | A) / [P(A) * P(D | A) + P(B) * P(D | B)]
Substitute the given values into the equation:
P(A | D) = 0.45 * 0.65 / [0.45 * 0.65 + 0.55 * 0.4] = 0.566.

Therefore, P(A | D) is 0.566.

To find the required probabilities, we'll use the given values and the tree diagram. Here's a breakdown of the steps:

(a) P(A) · P(D | A):
1. P(A) = x = 0.45 (given)
2. P(D | A) = r = 0.65 (given)
3. Multiply the two probabilities: P(A) · P(D | A) = 0.45 * 0.65 = 0.292.

Therefore, P(A) · P(D | A) = 0.292.

(b) P(B) · P(D | B):
1. P(B) = y = 0.55 (given)
2. P(D | B) = s = 0.35 (given)
3. Multiply the two probabilities: P(B) · P(D | B) = 0.55 * 0.35 = 0.1925.

Therefore, P(B) · P(D | B) = 0.1925.

(c) P(A | D):
1. P(A) = x = 0.45 (given)
2. P(D | A) = r = 0.65 (given)
3. P(D | not A) = t = 0.4 (given)
4. Apply Bayes' theorem:
P(A | D) = (P(A) · P(D | A)) / [(P(A) · P(D | A)) + (P(not A) · P(D | not A))]
P(not A) = 1 - P(A) = 1 - 0.45 = 0.55 (complement rule)
P(A | D) = (0.45 * 0.65) / [(0.45 * 0.65) + (0.55 * 0.4)]
P(A | D) ≈ 0.633.

Therefore, P(A | D) ≈ 0.633.

Note: Rounds in the final answers were done to three decimal places, as per the question's instructions.