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Chapter 5: Problem 68

In a January 2016 Harris Poll, each of 2252 American adults was asked the following question: "If you had to choose, which ONE of the following sports would you say is your favorite?" ("Pro Football is Still America's Favorite Sport," www.theharrispoll.com/sports/Americas_Fav_Sport_2016. html, retrieved April 25,2017 ). Of the survey participants, \(33 \%\) chose pro football as their favorite sport. The report also included the following statement, "Adults with household incomes of \(\$ 75,000-<\$ 100,000(48 \%)\) are especially likely to name pro football as their favorite sport, while love of this particular game is especially low among those in \(\$ 100,000+$$ households \)(21 \%)\( Suppose that the percentages from this poll are representative of American adults in general. Consider the following events: \)F=\( event that a randomly selected American adult names pro football as his or her favorite sport \)L=\( event that a randomly selected American has a household income of \)\$ 75,000-<\$ 100,000\( \)H=\( event that a randomly selected American has a household income of \)\$ 100,000+$$ b. Are the events \(F\) and \(L\) mutually exclusive? Justify your answer. c. Are the events \(H\) and \(L\) mutually exclusive? Justify your answer. d. Are the events \(F\) and \(H\) independent? Justify your answer.

Short Answer

Expert verified
In summary: - Events F and L (pro football favorite sport and household income of \(75,000-\)100,000) are not mutually exclusive, as they can occur simultaneously. - Events H and L (household income of \(100,000+) and \)household income of $75,000-<100,000) are mutually exclusive, as they cannot happen at the same time. - Events F and H (pro football favorite sport and household income of \(100,000+)) are independent, as P(F|H) = P(F).

Step by step solution

01

Analyze Event F and L (Mutually Exclusive or Not)

Event F represents that a randomly selected American adult names pro football as their favorite sport. Event L represents that a randomly selected American has a household income of \(75,000-\)100,000. These two events can occur simultaneously since a person can both like pro football and have a household income of \(75,000-\)100,000. Conclusion: Events F and L are not mutually exclusive.
02

Analyze Event H and L (Mutually Exclusive or Not)

Event H represents that a randomly selected American has a household income of \(100,000 or more, and Event L denotes a household income between \)75,000 and $100,000. These two events cannot happen simultaneously since a person cannot belong to both income ranges at the same time. Conclusion: Events H and L are mutually exclusive.
03

Analyze Event F and H (Independent or Not)

Event F denotes that a randomly selected American adult names pro football as their favorite sport, and event H represents a household income of $100,000 or more. To determine if they are independent, we want to check if P(F|H) = P(F). Given P(F) = 33%, P(H) = 21%, and the information in the problem, we can find P(F 鈭� H) as 21% * 33% = 6.93%. Now, we can check for independence: P(F|H) = P(F 鈭� H) / P(H) = (6.93%)/(21%)=~ 33%. Since P(F|H) = P(F), events F and H are independent. In conclusion: - Events F and L are not mutually exclusive. - Events H and L are mutually exclusive. - Events F and H are independent.

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Key Concepts

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

Mutually Exclusive Events
In statistics, the concept of mutually exclusive events is fundamental. Two events are considered mutually exclusive if they cannot happen simultaneously. This means there is no overlap between the two events. If one event occurs, the other cannot.

In the provided exercise, the analysis of events \(F\) and \(L\) shows that a person can both like pro football and have a household income of \(75,000-100,000\). Thus, \(F\) and \(L\) are not mutually exclusive, as they can occur together.

Conversely, when examining events \(H\) and \(L\), we see that one cannot belong to two different income brackets simultaneously. Therefore, \(H\) and \(L\) are mutually exclusive events. Remember, mutually exclusive events will have a probability that any two such events happening at the same time is zero. This is important when calculating probabilities, as it simplifies many problems, assuming that no other conditions or events are impacting these scenarios.
Independent Events
Independent events in probability have no influence on each other's outcomes. This means the occurrence of one event does not affect the likelihood of another event occurring.

To confirm independence, you compare the joint probability \(P(F \cap H)\) of the events happening together with the individual probability of each event. Specifically, for independent events, \(P(A | B) = P(A)\), where \(P(A | B)\) is the conditional probability of \(A\) given \(B\).

Using events \(F\) and \(H\) as an example: \(P(F) = 33\%\) and \(P(H) = 21\%\). Calculating the joint probability \(P(F \cap H) = 6.93\%\), the conditional probability \(P(F | H) = \frac{6.93\%}{21\%} \approx 33\%\) matches \(P(F)\). Consequently, \(F\) and \(H\) are independent events, as the condition holds true that knowing \(H\) occurred does not change the probability of \(F\) occurring. This principle is crucial when analyzing events where external factors or other events do not impact each other.
Probability
Probability is a measure of the likelihood that a particular event will occur. It's a fundamental concept in statistics that assists in predicting future outcomes and making informed decisions.

When determining the probability of events, a scale from \(0\) to \(1\) is used, where \(0\) indicates impossibility and \(1\) indicates certainty. Sometimes it's expressed as a percentage from \(0\%\) to \(100\%\).

Key probability rules include:
  • The sum of probabilities of all possible outcomes of a trial is \(1\).
  • For independent events \(A\) and \(B\), \(P(A \cap B) = P(A) \times P(B)\).
  • For mutually exclusive events \(A\) and \(B\), \(P(A \cup B) = P(A) + P(B)\).
Understanding these basic principles allows one to evaluate and predict the likelihood of various outcomes in different scenarios accurately. In the exercise, this understanding helps determine relationships between the given events \(F, L,\) and \(H\), guiding the correct usage of probability rules to assess mutual exclusivity and independence.

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Most popular questions from this chapter

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