/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 Suppose that an individual is ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 51

Suppose that an individual is randomly selected from the population of all adult males living in the United States. Let \(A\) be the event that the selected individual is over 6 feet in height, and let \(B\) be the event that the selected individual is a professional basketball player. Which do you think is greater, \(P(A \mid B)\) or \(P(B \mid A) ?\) Why?

Short Answer

Expert verified
The short answer is: \(P(A \mid B) > P(B \mid A)\). This is because the probability that an individual is over 6 feet tall given they are a professional basketball player is likely greater than the probability that an individual is a professional basketball player given they are over 6 feet tall. Most professional basketball players are over 6 feet tall, while only a small fraction of individuals over 6 feet tall are professional basketball players.

Step by step solution

01

Conditional Probability: \(P(A \mid B)\)

This represents the probability that an individual is over 6 feet tall, given that they are a professional basketball player. In other words, we are only looking at the population of professional basketball players and determining the likelihood that one is over 6 feet.
02

Conditional Probability: \(P(B \mid A)\)

This represents the probability that an individual is a professional basketball player, given that they are over 6 feet tall. In other words, we are only looking at the population of individuals over 6 feet tall and determining the likelihood that one of them is a professional basketball player. Now that we have a clear understanding of what each conditional probability represents, let's reason about which one is likely greater.
03

Comparing Conditional Probabilities

If we think about the population of professional basketball players, it is quite reasonable to assume that a high percentage of them are over 6 feet tall since height is an important factor in basketball. Therefore, \(P(A \mid B)\) is likely quite high. On the other hand, if we consider the population of individuals who are over 6 feet tall, only a very small fraction of them will be professional basketball players since there are many other professions and not everyone over 6 feet tall chooses to pursue basketball as a career. Therefore, \(P(B \mid A)\) is likely much smaller than \(P(A \mid B)\).
04

Conclusion

Based on our reasoning, it is more likely that \(P(A \mid B) > P(B \mid A)\). In other words, the probability that an individual is over 6 feet tall given they are a professional basketball player is greater than the probability that an individual is a professional basketball player given they are over 6 feet tall.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Theory
Probability theory is the mathematical framework that helps us quantify and analyze the likelihood of different events occurring. In simple terms, it allows us to make predictions about real-world situations using numbers between 0 and 1.
A probability of 0 means the event will not happen, while a probability of 1 indicates certainty.
Anything in between shows the degree of likelihood of an event happening.
  • For example, flipping a fair coin gives a probability of 0.5 for landing on heads.
  • Rolling a fair six-sided die gives a probability of 1/6 for any particular number.
Probability theory is crucial in many fields, such as statistics, finance, and computer science, where it helps analyze data and predict future trends.
Events and Outcomes
Events and outcomes are key components of probability theory. An event is a collection of results we are interested in, while an outcome is a possible result of an experiment or situation.
To understand better:
  • An "event" could be drawing a red card from a standard deck of cards.
  • An "outcome" could be drawing the king of hearts.
In our original exercise, the events are defined as follows:
  • Event A: An individual is over 6 feet tall.

  • Event B: An individual is a professional basketball player.

    • The outcomes are whether someone belongs to these groups or not. Understanding these terms helps us form conditional probabilities, allowing more nuanced analysis.
    Comparative Probability Analysis
    Comparative probability analysis involves comparing different probabilities to make informed conclusions.
    In the original problem, we are comparing two conditional probabilities: \(P(A \mid B)\) and \(P(B \mid A)\).
    • \(P(A \mid B)\): Probability an individual is over 6 feet, given they are a professional basketball player.
    • \(P(B \mid A)\): Probability an individual is a professional basketball player, given they are over 6 feet tall.
    By evaluating these, we can decide that \(P(A \mid B)\) is greater because, typically, basketball players are tall, while only a few tall people become professional players.
    This kind of analysis helps make sense of real-world situations by understanding how likely different scenarios are in comparison.

    One App. One Place for Learning.

    All the tools & learning materials you need for study success - in one app.

    Get started for free

    Most popular questions from this chapter

    A large cable company reports the following: \(80 \%\) of its customers subscribe to cable TV service \(42 \%\) of its customers subscribe to Internet service \(32 \%\) of its customers subscribe to telephone service \(25 \%\) of its customers subscribe to both cable TV and Internet service \(21 \%\) of its customers subscribe to both cable TV and phone service \(23 \%\) of its customers subscribe to both Internet and phone service \(15 \%\) of its customers subscribe to all three services Consider the chance experiment that consists of selecting one of the cable company customers at random. In Exercise \(5.53,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. \(P(\) cable TV only) b. \(P\) (Internet \(\mid\) cable TV) c. \(P(\) exactly two services \()\) d. \(P\) (Internet and cable TV only)

    A study of the impact of seeking a second opinion about a medical condition is described in the paper "Evaluation of Outcomes from a National Patient- Initiated Second-Opinion Program". Based on a review of 6791 patient-initiated second opinions, the paper states the following: "Second opinions often resulted in changes in diagnosis (14.8\%), treatment \((37.4 \%),\) or changes in both \((10.6 \%)\)." Consider the following two events: \(D=\) event that second opinion results in a change in diagnosis \(T=\) event that second opinion results in a change in treatment a. What are the values of \(P(D), P(T),\) and \(P(D \cap T) ?\) b. Use the given probability information to set up a hypothetical 1000 table with columns corresponding to \(D\) and \(D^{C}\) and rows corresponding to \(T\) and \(T^{C}\). c. What is the probability that a second opinion results in neither a change in diagnosis nor a change in treatment? d. What is the probability that a second opinion results is a change in diagnosis or a change in treatment?

    An airline reports that for a particular flight operating daily between Phoenix and Atlanta, the probability of an on-time arrival is 0.86. Give a relative frequency interpretation of this probability.

    5.65 The authors of the paper "Do Physicians Know When Their Diagnoses Are Correct?" (Journal of General Internal Medicine [2005]: 334-339) presented detailed case studies to medical students and to faculty at medical schools. Each participant was asked to provide a diagnosis in the case and also to indicate whether his or her confidence in the correctness of the diagnosis was high or low. Define the events \(C\), \(I,\) and \(H\) as follows: \(C=\) event that diagnosis is correct \(I=\) event that diagnosis is incorrect \(H=\) event that confidence in the correctness of the diagnosis is high a. Data appearing in the paper were used to estimate the following probabilities for medical students: $$ \begin{array}{r} P(C)=0.261 \\ P(I)=0.739 \\ P(H \mid C)=0.375 \\ P(H \mid I)=0.073 \end{array} $$ Use Bayes' Rule to calculate the probability of a correct diagnosis given that the student's confidence level in the correctness of the diagnosis is high. b. Data from the paper were also used to estimate the following probabilities for medical school faculty: $$ \begin{array}{r} P(C)=0.495 \\ P(I)=0.505 \\ P(H \mid C)=0.537 \\ P(H \mid I)=0.252 \end{array} $$ Calculate \(P(C \mid H)\) for medical school faculty. How does the value of this probability compare to the value of \(P(C \mid H)\) for students calculated in Part (a)?

    Phoenix is a hub for a large airline. Suppose that on a particular day, 8000 passengers arrived in Phoenix on this airline. Phoenix was the final destination for 1800 of these passengers. The others were all connecting to flights to other cities. On this particular day, several inbound flights were late, and 480 passengers missed their connecting flight. Of these 480 passengers, 75 were delayed overnight and had to spend the night in Phoenix. Consider the chance experiment of choosing a passenger at random from these 8000 passengers. Calculate the following probabilities: a. the probability that the selected passenger had Phoenix as a final destination. b. the probability that the selected passenger did not have Phoenix as a final destination. c. the probability that the selected passenger was connecting and missed the connecting flight. d. the probability that the selected passenger was a connecting passenger and did not miss the connecting flight. e. the probability that the selected passenger either had Phoenix as a final destination or was delayed overnight in Phoenix. f. An independent customer satisfaction survey is planned. Fifty passengers selected at random from the 8000 passengers who arrived in Phoenix on the day described above will be contacted for the survey. The airline knows that the survey results will not be favorable if too many people who were delayed overnight are included. Write a few sentences explaining whether or not you think the airline should be worried, using relevant probabilities to support your answer.
    See all solutions

    Recommended explanations on Math Textbooks

    Pure Maths

    Read Explanation

    Probability and Statistics

    Read Explanation

    Decision Maths

    Read Explanation

    Geometry

    Read Explanation

    Calculus

    Read Explanation

    Logic and Functions

    Read Explanation
    View all explanations

    What do you think about this solution?

    We value your feedback to improve our textbook solutions.

    Study anywhere. Anytime. Across all devices.