Create a Tree Diagram to model the scenario Question that requires application of Conditional Probability Question that requires the Law of Total Probability Question that implements Bayes’ Theorem via Conditional Probability

QUESTION

From your own experiences, create a scenario that will require the use of a tree diagram to determine probability of related events. Be sure that your scenario has at least two related events. (You may use the examples from section 8.7 of your text as a guide.) Create at least four questions based on your scenario:

• Create a Tree Diagram to model the scenario
• Question that requires application of Conditional Probability
• Question that requires the Law of Total Probability
• Question that implements Bayes’ Theorem via Conditional Probability

Answer each of your own posed questions, providing all related work.

ANSWER

Tree Diagram for Determining Probabilities in a Scenario

Introduction

In this scenario, we will explore the concept of using a tree diagram to determine the probability of related events. Tree diagrams are visual representations that help us understand the possible outcomes and probabilities of a series of interconnected events. Let’s consider a hypothetical scenario involving a customer’s purchase decision for a new smartphone. We will use this scenario to illustrate the application of conditional probability, the law of total probability, and Bayes’ Theorem.

Scenario:
A customer is considering purchasing a new smartphone from Brand X or Brand Y. The customer’s decision depends on two factors: the price of the phone and the availability of a specific feature, let’s say an advanced camera. The probabilities associated with each event are as follows:

1. The probability of choosing Brand X: P(X) = 0.6
2. The probability of choosing Brand Y: P(Y) = 0.4
3. The probability of having the advanced camera feature in Brand X: P(C|X) = 0.3
4. The probability of having the advanced camera feature in Brand Y: P(C|Y) = 0.6

Based on these probabilities, we can construct a tree diagram to visualize the different outcomes and their associated probabilities.

Tree Diagram:
“`
/ \
X Y
/ \ / \
C ~C C ~C
“`

The tree diagram represents the following possibilities:
1. Choosing Brand X and having the advanced camera feature (C)
2. Choosing Brand X and not having the advanced camera feature (~C)
3. Choosing Brand Y and having the advanced camera feature (C)
4. Choosing Brand Y and not having the advanced camera feature (~C)

Questions:

1. Question on Conditional Probability:
What is the probability that the customer chooses Brand X and has the advanced camera feature?
Solution:
To find this probability, we can use the concept of conditional probability:
P(X and C) = P(X) * P(C|X)
P(X and C) = 0.6 * 0.3 = 0.18

Therefore, the probability that the customer chooses Brand X and has the advanced camera feature is 0.18.

2. Question on the Law of Total Probability:
What is the probability that the customer chooses a phone with the advanced camera feature?
Solution:
To find this probability, we need to consider all possible outcomes:
P(C) = P(X and C) + P(Y and C)
= P(X) * P(C|X) + P(Y) * P(C|Y)
= 0.6 * 0.3 + 0.4 * 0.6
= 0.18 + 0.24
= 0.42

Therefore, the probability that the customer chooses a phone with the advanced camera feature is 0.42.

3. Question on Bayes’ Theorem via Conditional Probability:
Given that the customer has chosen a phone with the advanced camera feature, what is the probability that the customer chose Brand X?
Solution:
We can apply Bayes’ Theorem to find the probability:
P(X|C) = (P(C|X) * P(X)) / P(C)
= (0.3 * 0.6) / 0.42
= 0.18 / 0.42
= 0.4286 (approximately)

Therefore, the probability that the customer chose Brand X given that they have a phone with the advanced camera feature is approximately 0.4286.

Conclusion:
In this scenario, we used a tree diagram to model the probabilities of related events in a customer