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Chapter 6: Problem 5

A new model of laptop computer can be ordered with one of three screen sizes ( 10 inches, 12 inches, 15 inches) and one of four hard drive sizes \((50 \mathrm{~GB}, 100 \mathrm{~GB}\), \(150 \mathrm{~GB}\), and \(200 \mathrm{~GB}\) ). Consider the chance experiment in which a laptop order is selected and the screen size and hard drive size are recorded. a. Display possible outcomes using a tree diagram. b. Let \(A\) be the event that the order is for a laptop with a screen size of 12 inches or smaller. Let \(B\) be the event that the order is for a laptop with a hard drive size of at most \(100 \mathrm{~GB}\). What outcomes are in \(A^{C}\) ? In \(A \cup B ?\) In \(A \cap B\) ? c. Let \(C\) denote the event that the order is for a laptop with a \(200 \mathrm{~GB}\) hard drive. Are \(A\) and \(C\) disjoint events? Are \(B\) and \(C\) disjoint?

Short Answer

Expert verified
a. 12 outcomes represented in a tree diagram. b. \(A^{C}\): {(15, 50), (15, 100), (15, 150), (15, 200)}. \(A \cup B\): All 12 outcomes. \(A \cap B\): {(10, 50), (10, 100), (12, 50), (12, 100)}. c. A and C are not disjoint; B and C are disjoint.

Step by step solution

01

Construct a Tree Diagram

Start by drawing a tree diagram to represent all possibilities. The first branch indicates the screen size choices (10, 12, 15), and each of these branches further splits into four branches representing the hard drive choices (50, 100, 150, 200). This results in 12 outcomes overall.
02

Consider Event A and Its Complement

Event A consists of orders for laptops with a screen size of 12 inches or smaller, which is (10, 50), (10, 100), (10, 150), (10, 200), (12, 50), (12, 100), (12, 150), (12, 200). The complement of A, \(A^{C}\), consists of the remaining outcomes, which is (15, 50), (15, 100), (15, 150), (15, 200).
03

Consider Events A and B

Event B consists of orders for laptops with a hard drive size of at most 100 GB: (10, 50), (10, 100), (12, 50), (12, 100), (15, 50), (15, 100). The union of A and B, \(A \cup B\), is any outcome in either A or B or both, which includes all 12 outcomes. The intersection of A and B, \(A \cap B\), consists of outcomes present in both A and B, which are (10, 50), (10, 100), (12, 50), (12, 100).
04

Consider Events A, B, and C

Event C consists of orders for laptops with a 200 GB hard drive: (10, 200), (12, 200), (15, 200). A and C share two outcomes (10, 200) and (12, 200), so they are not disjoint. B and C do not share any outcomes, so they are disjoint.

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Key Concepts

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

Probability and Statistics
Probability and statistics are branches of mathematics that deal with data analysis and the likelihood of events occurring. A fundamental concept in this field is the representation of potential outcomes for experiments, which is where tools like tree diagrams come into play.

A tree diagram breaks down complex probability problems into simple, manageable parts. In our exercise, the tree diagram is used to represent all possible choices for screen sizes and hard drive capacities of a new laptop model. Each 'branch' of the diagram corresponds to one decision point, creating a visual representation of all combinations. This helps in making probability calculations more systematic and transparent.

Tree diagrams depict the sequential nature of decisions and their associated probabilities, if any, allowing us to explore and communicate the entire sample space of an experiment efficiently. By visualizing the outcomes, students can often more easily understand the relationship between events, such as the likelihood of ordering a laptop with a specific screen size or hard drive capacity.
Complement of an Event
In probability, the complement of an event is the set of all outcomes in the sample space that are not included in the event. For event A, denoted as 'the order is for a laptop with a screen size of 12 inches or smaller,' the complement is represented as \(A^{C}\) and includes all laptop configurations that do not meet the criteria of event A.

The concept of the complement is crucial because it allows us to calculate the probability of an event not occurring directly. In essence, if we know the probability of an event, we also know the probability of its complement since the sum of these probabilities equals one. In our exercise, we identified the outcomes for event A, thus easily deriving its complement, \(A^{C}\), as the remaining outcomes. Understanding complements is also foundational for grasping more complex concepts like the union and intersection of events.
Union and Intersection of Events
The union of two events, denoted by \( A \cup B \), consists of outcomes that are in either event A or event B or in both. It's the 'either/or' scenario in terms of probability, where we consider all possibilities that satisfy at least one of the event conditions. In our exercise, when we explore \( A \cup B \), we consider all laptops with a screen size of 12 inches or smaller, or with hard drive sizes of at most 100 GB, covering the entire set of outcomes.

Contrastingly, the intersection of two events, denoted by \( A \cap B \), is the set of outcomes that are in both event A and event B simultaneously. It鈥檚 the 'both' scenario where we focus on outcomes that meet both criteria. For the given exercise, \( A \cap B \) includes only laptops that are 12 inches or smaller with hard drives of at most 100 GB. Clarifying these concepts helps students think critically about how different conditions relate to each other and how to calculate probabilities that require multiple conditions to be satisfied at once.

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Most popular questions from this chapter

Consider the chance experiment in which both tennis racket head size and grip size are noted for a randomly selected customer at a particular store. The six possible outcomes (simple events) and their probabilities are displayed in the following table: $$ \begin{array}{llll} && {\text { Grip Size }} \\ \text { Head size } & 4_{8}^{\frac{3}{8}} \text { in. } & 4 \frac{1}{2} \text { in. } & 4 \frac{5}{8} \text { in. } \\ \hline \text { Midsize } & O_{1}(.10) & O_{2}(.20) & O_{3}(.15) \\ \text { Oversize } & O_{4}(.20) & O_{5}(.15) & O_{6}(.20) \\ \hline \end{array} $$ a. The probability that grip size is \(4 \frac{1}{2}\) inches (event \(A\) ) is \(P(A)=P\left(O_{2}\right.\) or \(\left.O_{5}\right)=.20+.15=.35\). How would you interpret this probability? b. Use the result of Part (a) to calculate the probability that grip size is not \(4 \frac{1}{2}\) inches c. What is the probability that the racket purchased has an oversize head (event \(B\) ), and how would you interpret this probability? d. What is the probability that grip size is at least \(4 \frac{1}{2}\) inches?

Five hundred first-year students at a state university were classified according to both high school GPA and whether they were on academic probation at the end of their first semester. The data are $$ \begin{array}{lcccc} && {\text { High School GPA }} \\ & 2.5 \text { to } & 3.0 \text { to } & 3.5 \text { and } & \\ \text { Probation } & <3.0 & <3.5 & \text { Above } & \text { Total } \\ \hline \text { Yes } & 50 & 55 & 30 & 135 \\ \text { No } & 45 & 135 & 185 & 365 \\ \text { Total } & 95 & 190 & 215 & 500 \\ \hline \end{array} $$ a. Construct a table of the estimated probabilities for each GPA-probation combination. b. Use the table constructed in Part (a) to approximate the probability that a randomly selected first-year student at this university will be on academic probation at the end of the first semester. c. What is the estimated probability that a randomly selected first-year student at this university had a high school GPA of \(3.5\) or above? d. Are the two outcomes selected student has a bigh school GPA of \(3.5\) or above and selected student is on academic probation at the end of the first semester independent outcomes? How can you tell? e. Estimate the proportion of first-year students with high school GPAs between \(2.5\) and \(3.0\) who are on academic probation at the end of the first semester. f. Estimate the proportion of those first-year students with high school GPAs \(3.5\) and above who are on academic probation at the end of the first semester.

A friend who works in a big city owns two cars, one small and one large. Three-quarters of the time he drives the small car to work, and one-quarter of the time he takes the large car. If he takes the small car, he usually has little trouble parking and so is at work on time with probability .9. If he takes the large car, he is on time to work with probability .6. Given that he was at work on time on a particular morning, what is the probability that he drove the small car?

Is ultrasound a reliable method for determining the gender of an unborn baby? The accompanying data on 1000 births are consistent with summary values that appeared in the online version of the Journal of Statistics Education ("New Approaches to Leaming Probability in the First Statistics Course" [2001]). $$ \begin{array}{ccc} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Female } \end{array} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Male } \end{array} \\ \hline \begin{array}{c} \text { Actual Gender Is } \\ \text { Female } \end{array} & 432 & 48 \\ \begin{array}{c} \text { Actual Gender Is } \\ \text { Male } \end{array} & 130 & 390 \\ \hline \end{array} $$ a. Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. b. Use the given information to estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male. c. Based on your answers to Parts (a) and (b), do you think that a prediction that a baby is male and a prediction that a baby is female are equally reliable? Explain.

Delayed diagnosis of cancer is a problem because it can delay the start of treatment. The paper "Causes of Physician Delay in the Diagnosis of Breast Cancer" (Archives of Internal Medicine \([2002]: 1343-1348)\) examined possible causes for delayed diagnosis for women with breast cancer. The accompanying table summarizes data on the initial written mammogram report (benign or suspicious) and whether or not diagnosis was delayed for 433 women with breast cancer. $$ \begin{array}{l|cc} & & \text { Diagnosis } \\ & \begin{array}{c} \text { Diagnosis } \\ \text { Delayed } \end{array} & \begin{array}{c} \text { Not } \\ \text { Delayed } \end{array} \\ \hline \begin{array}{l} \text { Mammogram Report Benign } \\ \text { Mammogram Report } \\ \text { Suspicious } \end{array} & 32 & 89 \\ & 8 & 304 \\ \hline \end{array} $$ Consider the following events: \(B=\) the event that the mammogram report says benign \(S=\) event that the mammogram report says suspicious \(D=\) event that diagnosis is delayed a. Assume that these data are representative of the larger group of all women with breast cancer. Use the data in the table to find and interpret the following probabilities: i. \(\quad P(B)\) ii. \(P(S)\) iii. \(P(D \mid B)\) iv. \(P(D \mid S)\) b. Remember that all of the 433 women in this study actually had breast cancer, so benign mammogram reports were, by definition, in error. Write a few sentences explaining whether this type of error in the reading of mammograms is related to delayed diagnosis of breast cancer.
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