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Chapter 5: Problem 6

A probability experiment is conducted in which the sample space of the experiment is, \(S=\\{1,2,3,4,5,6,7,8,9,10,11,12\\} .\) Let event \(E=\\{2,3,4,5,6,7\\},\) event \(F=\\{5,6,7,8,9\\},\) event \(G=\\{9,10,11,12\\},\) and event \(H=\\{2,3,4\\} .\) Assume each outcome is equally likely. List the outcomes in \(F\) and \(G .\) Are \(F\) and \(G\) mutually exclusive?

Short Answer

Expert verified
The outcomes in F and G are 9. F and G are not mutually exclusive.

Step by step solution

01

- Identify outcomes in F and G

The problem provides the events as sets: Event F: \(F=\{5,6,7,8,9\}\)Event G: \(G=\{9,10,11,12\}\).
02

- Determine the common outcomes (if any)

Check if there are any common elements between sets F and G. Compare each element in F with each element in G. Comparing the two sets, the number '9' is found in both events F and G.
03

- Are F and G mutually exclusive?

For two events to be mutually exclusive, they should have no outcomes in common.Since the numbers '9' are common in both events, F and G are not mutually exclusive.

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Key Concepts

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

Probability Experiment
A probability experiment is any process or action that leads to well-defined results, known as outcomes. Examples of probability experiments include rolling a die, flipping a coin, or drawing a card from a deck. In each of these experiments, we can clearly identify and count the possible results. For instance, rolling a six-sided die can result in any one of six outcomes: 1, 2, 3, 4, 5, or 6.
Sample Space
The sample space of a probability experiment is the set of all possible outcomes. It is usually denoted by S. For example, if a die is rolled, the sample space is {1, 2, 3, 4, 5, 6} because those are all the possible results. In the given exercise, the sample space is S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. This means the experiment being conducted can result in any one of these 12 numbers.
Outcomes
An outcome is a single result from a probability experiment. Each outcome is an element of the sample space. Outcomes can be grouped into events. For instance, in a die-rolling experiment, rolling a 2 is an outcome. In the provided exercise, specific events are defined as: E={2, 3, 4, 5, 6, 7}, F={5, 6, 7, 8, 9}, G={9, 10, 11, 12}, and H={2, 3, 4}. Each event consists of specific outcomes from the sample space S. With these definitions, the task was to find common outcomes between events F and G and determine if they were mutually exclusive. Since they share the outcome '9', they are not mutually exclusive.

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Most popular questions from this chapter

The following data represent the number of driver fatalities in the United States in 2002 by age for male and female drivers: $$\begin{array}{|l|c|c|} \hline \text { Age} & \text { Male } & \text { Female }\\\\\hline \text { Under } 16 & 228 & 108 \\\\\hline 16-20 & 5696 & 2386 \\\\\hline 21-34 & 13,553 & 4148 \\\\\hline 35-54 & 14,395 & 5017 \\\\\hline 55-69 & 4937 & 1708 \\\\\hline 70 \text { and over } & 3159 & 1529 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected driver fatality who was male was 16 to 20 years old? (b) What is the probability that a randomly selected driver fatality who was 16 to 20 was male? (c) Suppose you are a police officer called to the scene of a traffic accident with a fatality. The dispatcher states that the victim is 16 to 20 years old, but the gender is not known. Is the victim more likely to be male or female? Why?

A flush in the card game of poker occurs if a player gets five cards that are all the same suit (clubs, diamonds, hearts, or spades). Answer the following questions to obtain the probability of being dealt a flush in five cards. (a) We initially concentrate on one suit, say clubs. There are 13 clubs in a deck. Compute \(P\) (five clubs) \(=\) \(P(\) first card is clubs and second card is clubs and third card is clubs and fourth card is clubs and fifth card is clubs). (b) A flush can occur if we get five clubs or five diamonds or five hearts or five spades. Compute \(P\) (five clubs or five diamonds or five hearts or five spades). Note the events are mutually exclusive.

Companies whose stocks are listed on the NASDAQ stock exchange have their company name represented by either four or five letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NASDAQ?

The following data represent, in thousands, the type of health insurance coverage of people by age in the year 2002 $$\begin{array}{llllll}\hline & <18 & 18-44 & 45-64 & >64 \\\\\hline \text { Private } & 49,473 & 76,294 & 52,520 & 20,685 \\\\\hline \text { Government } & 19,662 & 11,922 & 9,227 & 32,813 \\\\\hline \text { None } & 8,531 & 25,678 & 9,106 & 258 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected individual who is less than 18 years old has no health insurance? (b) What is the probability that a randomly selected individual who has no health insurance is less than 18 years old?

On October \(5,2001,\) Barry Bonds broke Mark McGwire's home-run record for a single season by hitting his 71st and 72nd home runs. Bonds went on to hit one more home run before the season ended, for a total of \(73 .\) Of the 73 home runs, 24 went to right field, 26 went to right center field, 11 went to center field, 10 went to left center field, and 2 went to left field. (Source: Baseball-almanac.com) (a) What is the probability that a randomly selected home run was hit to right field? (b) What is the probability that a randomly selected home run was hit to left field? (c) Was it unusual for Barry Bonds to hit a home run to left field? Explain.
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