91Ó°ÊÓ

For customers purchasing a refrigerator at a certain appliance store, let \(A\) be the event that the refrigerator was manufactured in the U.S., \(B\) be the event that the refrigerator had an icemaker, and \(C\) be the event that the customer purchased an extended warranty. Relevant probabilities are $$ \begin{aligned} &P(A)=.75 \quad P(B \mid A)=.9 \quad P\left(B \mid A^{\prime}\right)=.8 \\ &P(C \mid A \cap B)=.8 \quad P\left(C \mid A \cap B^{\prime}\right)=.6 \\ &P\left(C \mid A^{\prime} \cap B\right)=.7 \quad P\left(C \mid A^{\prime} \cap B^{\prime}\right)=.3 \end{aligned} $$ a. Construct a tree diagram consisting of first-, second-, and third- generation branches, and place an event label and appropriate probability next to each branch. b. Compute \(P(A \cap B \cap C)\). c. Compute \(P(B \cap C)\). d. Compute \(P(C)\). e. Compute \(P(A \mid B \cap C)\), the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased.

Step by step solution

01

Introduction to Conditional Probability

We are given a series of probabilities that describe the relationships between events related to refrigerators. Our task will be to use these probabilities to calculate others, but first, we will draw a tree diagram to visualize these interactions.

02

Draw the Tree Diagram - First Generation

Start with the event \(A\), whether a refrigerator is manufactured in the U.S. The probabilities are \(P(A) = 0.75\) and \(P(A') = 0.25\).

03

Draw the Tree Diagram - Second Generation

For each \(A\) and \(A'\), add branches for the event \(B\), whether the refrigerator has an icemaker. The probabilities are \(P(B|A) = 0.9\) and \(P(B'|A) = 0.1\), and \(P(B|A') = 0.8\) and \(P(B'|A') = 0.2\).

04

Draw the Tree Diagram - Third Generation

For each \(A \cap B\), \(A \cap B'\), \(A' \cap B\), and \(A' \cap B'\), add branches for the event \(C\), whether an extended warranty is purchased. Use the given conditional probabilities: \(P(C|A \cap B) = 0.8\), \(P(C|A \cap B') = 0.6\), \(P(C|A' \cap B) = 0.7\), \(P(C|A' \cap B') = 0.3\).

05

Calculate P(A ∩ B ∩ C)

Use the chain rule of probability along the path \(A \cap B \cap C\). \[P(A \cap B \cap C) = P(A) \cdot P(B|A) \cdot P(C|A \cap B) = 0.75 \times 0.9 \times 0.8 = 0.54\]

06

Calculate P(B ∩ C)

Use the total probability rule. Consider both scenarios \(A\) and \(A'\): \[P(B \cap C) = P(A \cap B \cap C) + P(A' \cap B \cap C)\] For \(P(A' \cap B \cap C)\), calculate: \[P(A') \cdot P(B|A') \cdot P(C|A' \cap B) = 0.25 \times 0.8 \times 0.7 = 0.14\] Finally, add the probabilities: \[P(B \cap C) = 0.54 + 0.14 = 0.68\]

07

Calculate P(C)

Consider all scenarios for \(C\): \[P(C) = P(A \cap B \cap C) + P(A \cap B' \cap C) + P(A' \cap B \cap C) + P(A' \cap B' \cap C)\] Compute first for each scenario: \[P(A \cap B' \cap C) = 0.75 \cdot 0.1 \cdot 0.6 = 0.045\] \[P(A' \cap B' \cap C) = 0.25 \cdot 0.2 \cdot 0.3 = 0.015\] Add all probabilities: \[P(C) = 0.54 + 0.045 + 0.14 + 0.015 = 0.74\]

08

Calculate P(A | B ∩ C)

Use Bayes' Theorem: \[P(A | B \cap C) = \frac{P(A \cap B \cap C)}{P(B \cap C)}\] Use previously calculated values: \[\frac{0.54}{0.68} \approx 0.794\]

09

Conclusion

Based on the calculations, we obtained the required probabilities using a tree diagram and the rules of probability. Make sure to check each calculation along the tree to ensure accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tree Diagram

A tree diagram is a foundational tool in probability that helps visualize the process of determining the likelihood of various combinations of events. Imagine a tree, with branches representing choices or events. Each split in the branches showcases two or more possibilities, effectively mapping out all potential outcomes of the sequence of events.
For example, our tree starts with the probability of a refrigerator being manufactured in the U.S. (event A). From there, it branches to include whether the refrigerator has an icemaker (event B). Finally, it branches to show the purchase of an extended warranty (event C).
Tree diagrams are crucial because they organize complex scenarios into manageable steps, revealing the chain of events and their associated probabilities as per provided data.

Chain Rule of Probability

The chain rule of probability is essential when we want to find the combination of multiple events occurring in sequence. It allows us to calculate the joint probability of events by breaking it down into conditional probabilities.
In our example, to calculate the probability of all three events occurring—manufacturing in the U.S. (A), having an icemaker (B), and purchasing an extended warranty (C)—the chain rule utilizes each step's probabilities:

• First, the probability of A (U.S. made refrigerator).
• Then, the conditional probability of B given A.
• Finally, the conditional probability of C given A and B.

The formula that reflects this is: \[P(A \cap B \cap C) = P(A) \cdot P(B|A) \cdot P(C|A \cap B)\]
This approach effectively breaks down the complexity into understandable segments, helping to manage multiple probabilities in sequence.

Total Probability Rule

The total probability rule helps in determining the overall likelihood of an event by accounting for all possible pathways leading to that event. When faced with a scenario where multiple conditions or events can lead to a particular outcome, this rule is applied.
For instance, to calculate the probability of purchasing a refrigerator with an icemaker and an extended warranty (B & C), we need to consider both possibilities: whether the fridge is U.S. manufactured (A) or not (A'). The rule can be summed up as:

• Path through A.
• Path through A'.

It’s calculated as: \[P(B \cap C) = P(A \cap B \cap C) + P(A' \cap B \cap C)\]
Each component is determined and summed to provide the total probability, thus offering a comprehensive view of all possible scenarios.

Bayes' Theorem

Bayes' theorem is a powerful tool in probability that allows us to update our beliefs about an event based on new information. Originally, it was used to flip conditional probabilities around to find out how likely a cause is given that we have seen an effect.
In our example scenario, we determine the likelihood that a refrigerator is made in the U.S., given that both an icemaker and an extended warranty were purchased (A | B & C). This requires applying Bayes' theorem, which is formulated as: \[P(A | B \cap C) = \frac{P(A \cap B \cap C)}{P(B \cap C)}\]
By using this theorem, we can reverse the condition and calculate the probability of a preliminary event based on the final outcome. This form of inference is pivotal in decision-making and updating our understanding of probability.