/*! 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 32 Suppose that an individual is ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 32

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 oyer 6 ft in height, and let \(B\) be the event that the selected individual is a professional basketball player. Which do you think is larger, \(P(A \mid B)\) or \(P(B \mid A) ?\) Why?

Short Answer

Expert verified
The probability of an adult male being over 6 ft in height given that he is a professional basketball player (\(P(A|B)\)) is more likely to be larger than the probability of an adult male being a professional basketball player given that he is over 6 ft (\(P(B|A)\)).

Step by step solution

01

Understanding the Concept of Conditional Probability

Conditional probability is the probability of an event given that another event has occurred. In this case, \(P(A|B)\) denotes the probability of 'A' happening given that 'B' has already happened. Similarly, \(P(B|A)\) denotes the probability of 'B' happening given that 'A' has already happened.
02

Relate to Real-world Scenario

In the context of the problem, \(P(A|B)\) refers to the probability of an adult male being over 6 ft in height given that he is already confirmed to be a professional basketball player. This is likely to be high because the majority of professional basketball players are tall. On the other hand, \(P(B|A)\) refers to the probability that an adult male is a professional basketball player given that he is known to be over 6ft. While many professional basketball players are indeed tall, not all tall individuals are professional basketball players.
03

Conclusion

Therefore, it is highly probable to suggest that \(P(A|B)\) is higher than \(P(B|A)\) because the probability of a professional basketball player being over 6ft is higher than the probability of a tall person being a professional basketball player.

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 a mathematical framework used to understand and quantify uncertainty. This branch of mathematics considers the likelihood of events happening. The cornerstone of probability theory is the concept of an event, which can be any outcome or combination of outcomes from an experiment or situation.
  • Events: An event is a set of outcomes to which a probability is assigned. For example, flipping a coin can result in two possible events: heads or tails.
  • Probability: Probability is the measure of the likelihood that an event will occur. It ranges from 0 (the event will not happen) to 1 (the event will certainly happen).
In conditional probabilities, such as those involved in the initial problem, the likelihood of an event is calculated based on the fact that another event has already occurred. This is crucial in many real-world applications, as it allows one to better predict very specific scenarios, helping in various fields such as meteorology, finance, and sports analytics.
Event Probability
Event probability involves calculating the chance of a particular event occurring. This exercises one鈥檚 understanding of probability within specific contexts.
The basic formula for event probability is given by:
  • If A is an event, the probability of A happening, denoted by P(A), is calculated as: \ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \.
Conditional probability further extends this idea by asking for the probability of an event happening, given that another event has already occurred, denoted by \( P(A|B)\).Let's look closer at the conditional probabilities in the initial problem:
  • \( P(A|B)\) involves knowing someone is a professional basketball player and then asking if they are over 6ft tall. Given the requirements and typical height of professional basketball players, this is generally a high probability.
  • \( P(B|A)\) involves knowing someone is over 6ft tall and asking whether they are a professional basketball player. This probability is low because there are many tall individuals who do not play professional basketball.
Real-world Application
Probability is not just a theoretical construct. It has practical applications in everyday life. Understanding and calculating probabilities can help in making better decisions based on available data.Consider the context provided in the exercise:
  • Hiring decisions in sports: For example, when selecting players for basketball teams, scouts often consider conditional probabilities related to height and playing skill, similar to \( P(A|B)\) from the exercise.
  • Marketing strategies: Companies might use probability to decide how likely a person who has purchased a sports-related product will buy again.
By understanding conditional probabilities, businesses can make informed predictions about market trends and consumer behavior, thus optimizing their marketing campaigns and operations.In summary, conditional probability is an invaluable tool in analyzing scenarios where multiple factors influence outcomes, providing insights into unpredictable and complex systems.

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

An article in the New York Times (March 2, 1994) reported that people who suffer cardiac arrest in New York City have only a 1 in 100 chance of survival. Using probability notation, an equivalent statement would be \(P\) (survival \()=.01\) for people who suffer a cardiac arrest in New York City. (The article attributed this poor survival rate to factors common in large cities: traffic congestion and the difficulty of finding victims in large buildings.) a. Give a relative frequency interpretation of the given probability. b. The research that was the basis for the New York Times article was a study of 2329 consecutive cardiac arrests in New York City. To justify the " 1 in 100 chance of survival" statement, how many of the 2329 cardiac arrest sufferers do you think survived? Explain.

Consider a system consisting of four components, as pictured in the following diagram: Components 1 and 2 form a series subsystem, as do Components 3 and 4 . The two subsystems are connected in parallel. Suppose that \(P(1\) works \()=.9, P(2\) works \()=.9\), \(P(3\) works \()=.9\), and \(P(4\) works \()=.9\) and that the four components work independently of one another. a. The \(1-2\) subsystem works only if both components work. What is the probability of this happening? b. What is the probability that the \(1-2\) subsystem doesn't work? that the \(3-4\) subsystem doesn't work? c. The system won't work if the \(1-2\) subsystem doesn't work and if the \(3-4\) subsystem also doesn't work. What is the probability that the system won't work? that it will work? d. How would the probability of the system working change if a \(5-6\) subsystem were added in parallel with the other two subsystems? e. How would the probability that the system works change if there were three components in series in each of the two subsystems?

The Cedar Rapids Gazette (November 20, 1999) reported the following information on compliance with child restraint laws for cities in Iowa: $$ \begin{array}{lcc} & \begin{array}{c} \text { Number of } \\ \text { Children } \\ \text { Observed } \end{array} & \begin{array}{c} \text { Number } \\ \text { Properly } \\ \text { City } \end{array} & \text { Restrained } \\ \hline \text { Cedar Falls } & 210 & 173 \\ \text { Cedar Rapids } & 231 & 206 \\ \text { Dubuque } & 182 & 135 \\ \text { Iowa City (city) } & 175 & 140 \\ \text { Iowa City (interstate) } & 63 & 47 \\ \hline \end{array} $$ a. Use the information provided to estimate the following probabilities: i. The probability that a randomly selected child is properly restrained given that the child is observed in Dubuque. ii. The probability that a randomly selected child is properly restrained given that the child is observed in a city that has "Cedar" in its name. b. Suppose that you are observing children in the Iowa City area. Use a tree diagram to illustrate the possible outcomes of an observation that considers both the location of the observation (city or interstate) and whether the child observed was properly restrained.

The newspaper article "Folic Acid Might Reduce Risk of Down Syndrome" (USA Today, September 29 , 1999) makes the following statement: "Older women are at a greater risk of giving birth to a baby with Down Syndrome than are younger women. But younger women are more fertile, so most children with Down Syndrome are born to mothers under \(30 .\) " Let \(D=\) event that a randomly selected baby is born with Down Syndrome and \(Y=\) event that a randomly selected baby is born to a young mother (under age 30 ). For each of the following probability statements, indicate whether the statement is consistent with the quote from the article, and if not, explain why not. a. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) b. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.7\) c. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) d. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\) e. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.4\) f. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\)

A student placement center has requests from five students for interviews regarding employment with a particular consulting firm. Three of these students are math majors, and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students; these two will be randomly selected from among the five. a. What is the probability that both selected students are statistics majors? b. What is the probability that both students are math majors? c. What is the probability that at least one of the students selected is a statistics major? d. What is the probability that the selected students have different majors?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.