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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.

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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.

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    Most popular questions from this chapter

    An online store offers two methods of shipping-regular ground service and an expedited 2 -day shipping. Customers may also choose whether or not to have the purchase gift wrapped. Suppose that the events \(E=\) event that the customer chooses expedited shipping \(G=\) event that the customer chooses gift wrap are independent with \(P(E)=0.26\) and \(P(G)=0.12\). a. Construct a hypothetical 1000 table with columns corresponding to whether or not expedited shipping was chosen and rows corresponding to whether or not gift wrap was selected. b. Use the information in the table to calculate \(P(E \cup G)\). Give a long- run relative frequency interpretation of this probability.

    The article "Scrambled Statistics: What Are the Chances of Finding Multi-Yolk Eggs?" (Significance [August 2016]: 11) gives the probability of a double-yolk egg as 0.001 . a. Give a relative frequency interpretation of this probability. b. If 5000 eggs were randomly selected, about how many double-yolk eggs would you expect to find?

    Suppose that \(20 \%\) of all teenage drivers in a certain county received a citation for a moving violation within the past year. Assume in addition that \(80 \%\) of those receiving such a citation attended traffic school so that the citation would not appear on their permanent driving record. Consider the chance experiment that consists of randomly selecting a teenage driver from this county. a. One of the percentages given in the problem specifies an unconditional probability, and the other percentage specifies a conditional probability. Which one is the conditional probability, and how can you tell? b. Suppose that two events \(E\) and \(F\) are defined as follows: \(E=\) selected driver attended traffic school \(F=\) selected driver received such a citation Use probability notation to translate the given information into two probability statements of the form \(P(\underline{C})=\) probability value.

    A man 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 0.9. If he takes the large car, he is on time to work with probability 0.6. Given that he was at work on time on a particular morning, what is the probability that he drove the small car?

    5.57 There are two traffic lights on Shelly's route from home to work. Let \(E\) denote the event that Shelly must stop at the first light, and define the event \(F\) in a similar manner for the second light. Suppose that \(P(E)=0.4, P(F)=0.3\) and \(P(E \cap F)=0.15 .\) In Exercise \(5.25,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. The probability that Shelly must stop for at least one light (the probability of the event \(E \cup F\) ). b. The probability that Shelly does not have to stop at either light. c. The probability that Shelly must stop at exactly one of the two lights. d. The probability that Shelly must stop only at the first light.
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