Please help with how to work this problem. The image displays a tree diagram that represents a two-stage experiment. The diagram consists of three branches (A, B, C) departing from an initial point. Each branch further splits into two sub-branches, resulting in outcomes $ D $ and $ D^c $.### Detailed Breakdown of the Diagram- **From the initial point:** - **Branch A:** - Probability to A: $ \frac{1}{5} $ - Splits into: - $ D $ with probability $ \frac{1}{2} $ - $ D^c $ with probability $ \frac{1}{2} $ - **Branch B:** - Probability to B: $ \frac{2}{5} $ - Splits into: - $ D $ with probability $ \frac{1}{3} $ - $ D^c $ with probability $ \frac{2}{3} $ - **Branch C:** - Probability to C: $ \frac{2}{5} $ - Splits into: - $ D $ with probability $ \frac{7}{8} $ - $ D^c $ with probability $ \frac{1}{8} $### TasksUse this diagram to find the following probabilities:(a) $ P(A \cap D) $(b) $ P(B \cap D) $(c) $ P(C \cap D) $(d) $ P(D) $(e) Verify with the equation:\[P(A \cap D) + P(B \cap D) + P(C \cap D) = P(A) \cdot P(D \mid A) + P(B) \cdot P(D \mid B) + P(C) \cdot P(D \mid C)\] **Description:**This image contains a series of probability equations and a diagram used to calculate those probabilities. The diagram is a probability tree, typically used to portray different branches and possible outcomes of an event. Here's an explanation of the content and layout:---**Instructions:**Use this diagram to find the following probabilities.**Questions:**(a) $ P(A \cap D) $ [Text Box](b) $ P(B \cap D) $ [Text Box](c) $ P(C \cap D) $ [Text Box](d) $ P(D) $ [Text Box](e) **Verify:**\[ P(A | D) = \frac{P(A \cap D)}{P(D)} = \frac{P(A) \cdot P(D | A)}{P(A) \cdot P(D | A) + P(B) \cdot P(D | B) + P(C) \cdot P(D | C)} \]--- **Calculation Fields:**The equation is broken down into components, with empty fields provided for the calculation of specific probability values. These fields include probability intersections and conditional probabilities.1. $ P(A \cap D) $: [Text Box]2. Substitution and calculation within the formula: [Multiple Text Boxes]3. Final verification fields for resulting expressions: [Text Box]The task involves calculating specific probabilities based on given probabilities in a tree diagram and verifying them using a given formula.

Name: Please help with how to work this problem. The image displays a tree d
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Description: Please help with how to work this problem. The image displays a tree diagram that represents a two-stage experiment. The diagram consists of three branches (A, B, C) departing from an initial point. Each branch further splits into two sub-branches, r

The values given on the edges of the diagram are theprobabilities. From the diagram we can determine P (A) = 15, P (B) = 25, P(C) = 25 P (D/A) = 12, P (D/B) = 13, P (D/C) = 78(a). P (A∩D) = P (A)×P (D/A) = 15×12 = 110 (b). P (B∩D) = P (B)×P (D/B) = 25×13 = 215 (c). P (C∩D) = P (C)×P (D/C) = 25×78 = 720 (d). P (D) = P (A∩D) + P (B∩D) + P (C∩D) = P (A)×P (D/A) + P (B)×P (D/B) + P (C)×P (D/C) = 15×12 + 25×13 + 25×78 = 110 + 215 + 720 = 1360 = 0.2167 (e). P (A/D) = P (A∩D)P (D) = P(A) P(D/A)P(A) P(D/A) + P(B) P(D/B) + P(C) P(D/C) ⇒ 1101360 = 15×1215×12+25×13+25×78 ⇒ 613 = 110110+215+720 ⇒ 613 = 1101360 ⇒ 613 = 613 ⇒ L.H.S = R.H.S Therefore, the result is verified

Math

Probability

The accompanying tree diagram represents a two-stage experiment. 1/2 1/5 1/2 -D° 1/3 2/5 B 2/3 D° D 2/5 7/8 1/8 D° Use this diagram to find the following probabilities. (a) P(A N D) (b) P(B N D) (c) P(C N D) (d) P(D) (e) Verify. P(AN D) P(D) P(A | D) = P(A) P(D | A) P(A) P(D | A) + P(B) · P(D | B) + P(C) • P(D | C)

The accompanying tree diagram represents a two-stage experiment. 1/2 1/5 1/2 -D° 1/3 2/5 B 2/3 D° D 2/5 7/8 1/8 D° Use this diagram to find the following probabilities. (a) P(A N D) (b) P(B N D) (c) P(C N D) (d) P(D) (e) Verify. P(AN D) P(D) P(A | D) = P(A) P(D | A) P(A) P(D | A) + P(B) · P(D | B) + P(C) • P(D | C)

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Please help with how to work this problem.

The image displays a tree diagram that represents a two-stage experiment. The diagram consists of three branches (A, B, C) departing from an initial point. Each branch further splits into two sub-branches, resulting in outcomes $ D $ and $ D^c $.

### Detailed Breakdown of the Diagram

- **From the initial point:**
- **Branch A:**
- Probability to A: $ \frac{1}{5} $
- Splits into:
- $ D $ with probability $ \frac{1}{2} $
- $ D^c $ with probability $ \frac{1}{2} $
- **Branch B:**
- Probability to B: $ \frac{2}{5} $
- Splits into:
- $ D $ with probability $ \frac{1}{3} $
- $ D^c $ with probability $ \frac{2}{3} $
- **Branch C:**
- Probability to C: $ \frac{2}{5} $
- Splits into:
- $ D $ with probability $ \frac{7}{8} $
- $ D^c $ with probability $ \frac{1}{8} $

### Tasks

Use this diagram to find the following probabilities:

(a) $ P(A \cap D) $

(b) $ P(B \cap D) $

(c) $ P(C \cap D) $

(d) $ P(D) $

(e) Verify with the equation:

\[
P(A \cap D) + P(B \cap D) + P(C \cap D) = P(A) \cdot P(D \mid A) + P(B) \cdot P(D \mid B) + P(C) \cdot P(D \mid C)
\]

**Description:**

This image contains a series of probability equations and a diagram used to calculate those probabilities. The diagram is a probability tree, typically used to portray different branches and possible outcomes of an event. Here's an explanation of the content and layout:

---

**Instructions:**

Use this diagram to find the following probabilities.

**Questions:**

(a) $ P(A \cap D) $
[Text Box]

(b) $ P(B \cap D) $
[Text Box]

(c) $ P(C \cap D) $
[Text Box]

(d) $ P(D) $
[Text Box]

(e) **Verify:**

\[
P(A | D) = \frac{P(A \cap D)}{P(D)} = \frac{P(A) \cdot P(D | A)}{P(A) \cdot P(D | A) + P(B) \cdot P(D | B) + P(C) \cdot P(D | C)}
\]

---

**Calculation Fields:**

The equation is broken down into components, with empty fields provided for the calculation of specific probability values. These fields include probability intersections and conditional probabilities.

1. $ P(A \cap D) $: [Text Box]
2. Substitution and calculation within the formula: [Multiple Text Boxes]
3. Final verification fields for resulting expressions: [Text Box]

The task involves calculating specific probabilities based on given probabilities in a tree diagram and verifying them using a given formula.

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