91Ó°ÊÓ

The three most popular options on a certain type of new car are a built-in GPS \((A)\), a sunroof \((B)\), and an automatic transmission \((C)\). If \(40 \%\) of all purchasers request \(A, 55 \%\) request \(B, 70 \%\) request \(C, 63 \%\) request \(A\) or \(B\), \(77 \%\) request \(A\) or \(C, 80 \%\) request \(B\) or \(C\), and \(85 \%\) request \(A\) or \(B\) or \(C\), determine the probabilities of the following events. [Hint: " \(A\) or \(B\) " is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.] a. The next purchaser will request at least one of the three options. b. The next purchaser will select none of the three options. c. The next purchaser will request only an automatic transmission and not either of the other two options. d. The next purchaser will select exactly one of these three options.

Step by step solution

01

Understand Given Information

We are given percentages for different combinations of requests for three car options: GPS \(A\), Sunroof \(B\), and Automatic Transmission \(C\). We must determine probabilities for certain requests.

02

Draw a Venn Diagram and Label Regions

Draw a circle for each of the options A, B, and C. Label the union of the circles according to the given percentages: \( P(A) = 0.40 \), \( P(B) = 0.55 \), \( P(C) = 0.70 \), \( P(A \cup B) = 0.63 \), \( P(A \cup C) = 0.77 \), \( P(B \cup C) = 0.80 \), \( P(A \cup B \cup C) = 0.85 \).

03

Calculate Overlapping Areas Using Inclusion-Exclusion Principle

Use the principle \( P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) \). Substitute known values to find the intersections: \( P(A \cap B) \), \( P(A \cap C) \), and \( P(B \cap C) \).

04

Solve for Each Intersection

From \( P(A \cup B) = 0.63 = 0.40 + 0.55 - P(A \cap B) \), solve for \( P(A \cap B) = 0.32 \). Similarly, solve for other intersections using \( P(A \cup C) = 0.77 \) and \( P(B \cup C) = 0.80 \).

05

Calculate \( P(A \cap B \cap C) \)

Use \( P(A \cup B \cup C) = 0.85 = 0.40 + 0.55 + 0.70 - 0.32 - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) \) to find \( P(A \cap B \cap C) \).

06

Determine Probability for Each Part

a. \( P(A \cup B \cup C) = 0.85 \), which means 85% will request at least one option. b. Probability that none is requested is the complement: \( 1 - P(A \cup B \cup C) = 0.15 \).c. To request only C, use \( P(C) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) \). d. For selecting exactly one, calculate individually: \( P(A \cap \overline{B} \cap \overline{C}) \), \( P(B \cap \overline{A} \cap \overline{C}) \), and \( P(C \cap \overline{A} \cap \overline{B}) \).

07

Verify Each Probability Calculation

Ensure each calculation aligns with the total probability being 100% (or 1 in decimal form) for events listed a through d as per conditions.

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Key Concepts

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

Venn Diagram

A Venn Diagram is a helpful tool used to visualize relationships between different sets or events. In our problem, we have three main events: requests for built-in GPS \((A)\), a sunroof \((B)\), and an automatic transmission \((C)\). By drawing circles to represent each event, we can easily see how they overlap and where they stand individually.

In this exercise, a Venn Diagram can help us understand various combinations of these options. For example, the area where all three circles overlap represents customers who choose all three options. The unique segments of each circle that don't overlap with others show customers selecting only that option. Using a Venn Diagram simplifies the complexity of multiple overlapping requests. It provides a clear visual depiction to determine probabilities based on data for each option and their combinations.

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is crucial when calculating the probability of unions of multiple events. It helps in finding the exact probability of one or more events occurring by accounting for overlapping probabilities.

In our example problem, with events \(A\), \(B\), and \(C\), the principle is used to prevent double-counting of these overlaps. The principle states that:

• \( P(A \cup B \cup C) = P(A) + P(B) + P(C) \)
• Subtract intersections of pairs: \(- P(A \cap B) - P(A \cap C) - P(B \cap C) \)
• Add the intersection of all three \(+ P(A \cap B \cap C) \) to compensate for removing it thrice.

Depending on these calculations and given percentages, we can determine exact individual probabilities of events and their interactions, reinforcing system integrity and accuracy.

Complement Rule

The complement rule is a fundamental concept in probability theory which assists in finding the probability of the opposite or 'complement' of an event occurring. It is defined as:

• If \(A\) is an event, then its complement \(\overline{A}\) represents scenarios where \(A\) does not occur.
• The probability of \(\overline{A}\) is \( P(\overline{A}) = 1 - P(A) \).

In relation to our exercise, to determine the probability that none of the car options is selected, we use the complement rule on the probability of at least one option being chosen \(P(A \cup B \cup C)\). Thus, the required probability becomes \(1 - P(A \cup B \cup C) = 0.15\). This makes solving such problems straightforward and efficient by focusing on what doesn't happen.

Intersection of Events

The intersection of events focuses on scenarios where two or more specific events occur simultaneously. In mathematical terms, it is the probability that event \(A\) and \(B\) happen together, denoted by \(P(A \cap B)\).

For our problem's context, intersection probabilities like \(P(A \cap B)\), \(P(A \cap C)\), and \(P(B \cap C)\) indicate customers requesting any two options concurrently.

We calculated these intersections using given percentages and the inclusion-exclusion principle, helping us understand overlaps—providing insights into combinations often requested. By identifying intersections, we can also address specific scenarios like a customer choosing only one option, further highlighting the utility of determining intersections in probability analysis.