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Chapter 7: Problem 16

Let \(S\) be any sample space and let \(E_{E}\) \(\boldsymbol{F}_{\boldsymbol{r}}\) and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(U, \cap\), and '. The event that both \(E\) and \(F\) occur

Short Answer

Expert verified
The event that both \(E\) and \(F\) occur is represented by the intersection operation in set theory notation, which is denoted as \(E \cap F\).

Step by step solution

01

Interpret the Problem Statement

The task requires to describe the event that both \(E\) and \(F\) occur using set operations symbols. In set theory, the intersection operation (\(\cap\)) is used to denote the event where both of two events occur.
02

Use Intersection Operation

To depict the event that both \(E\) and \(F\) occur, we use the intersection symbol \(\cap\). Thus, the event that both \(E\) and \(F\) occur is represented as: \[ E \cap F \] in set theory notation. This denotes all outcomes that are common to both \(E\) and \(F\), i.e., all outcomes for which both events \(E\) and \(F\) occur.

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Key Concepts

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

Sample Space
A sample space, often denoted as \( S \), is a fundamental concept in probability theory. It is the set of all possible outcomes that can result from a particular experiment or random trial. Understanding the sample space helps in identifying all potential scenarios in a given problem, which is crucial for probability calculations.
  • Every probability problem starts with defining the sample space.
  • Each element in the sample space is a unique possible outcome.
  • The sample space must be exhaustive, meaning it includes all potential outcomes of an experiment.
Take, for example, the flip of a coin. Here, the sample space \( S \) consists of "Heads" and "Tails," expressed as \( \{ H, T \} \). In contrast, for a six-sided die, the sample space is \( \{ 1, 2, 3, 4, 5, 6 \} \). Identifying each possible outcome allows us to visualize and manipulate how events relate to one another within this set.
Intersection of Events
The intersection of events is a key concept that helps in finding the probability of multiple events occurring simultaneously. It is represented by the symbol \( \cap \) and highlights the outcomes common to two or more events. Let's dive deeper into the concept of intersection using the events \( E \) and \( F \).To denote the situation where both events \( E \) and \( F \) happen:
  • The intersection of \( E \) and \( F \) is shown as \( E \cap F \).
  • This describes the set of all outcomes that belong to both \( E \) and \( F \).
  • It is sometimes referred to as "taking both events together" or the "common part" of the events.
Understanding intersections is important for many problems in probability and statistics. Consider an example of rolling a die and getting an even number and a number greater than 2. The intersection would only include the numbers \( \{4, 6\} \), as these satisfy both conditions.
Set Theory Notation
Set theory notation is the language used to describe and analyze collections of objects. In probability, it is employed to articulate the relationships between various events and their outcomes. Some of the primary symbols in set theory notation include:
  • \( \cap \): Represents the intersection of sets, where outcomes common to both sets are included.
  • \( \cup \): Denotes the union of sets, meaning all outcomes from either or both sets are taken.
  • \( \complement \) or \( ' \): Indicates the complement of a set, which contains all outcomes not in the specified set.
For the exercise where both \( E \) and \( F \) occur, we use the intersection symbol as \( E \cap F \) to depict this scenario. This notation not only simplifies expression but also aids in the clear visualization of how events interact. Mastery of set theory notation is crucial as it underpins more complex operations and concepts within mathematics and probability theory.

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Most popular questions from this chapter

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or \(7 ?\) Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?

According to the Centers for Disease Control and Prevention, the percentage of adults \(25 \mathrm{yr}\) and older who smoke, by educational level, is as follows: $$\begin{array}{lcccccc} \hline & & & \text { High } & {\text { Under- }} \\ \text { Educational } & \text { No } & \text { GED } & \text { school } & \text { Some } & \text { graduate } & \text { Graduate } \\ \text { Level } & \text { diploma } & \text { diploma } & \text { graduate } & \text { college } & \text { level } & \text { degree } \\ \hline \text { Respondents, \% } & 26 & 43 & 25 & 23 & 10.7 & 7 \\ \hline \end{array}$$ In a group of 140 people, there were 8 with no diploma, 14 with GED diplomas, 40 high school graduates, 24 with some college, 42 with an undergraduate degree, and 12 with a graduate degree. (Assume that these categories are mutually exclusive.) If a person selected at random from this group was a smoker, what is the probability that he or she is a person with a graduate degree?

There were 80 male guests at a party. The number of men in each of four age categories is given in the following table. The table also gives the probability that a man in the respective age category will keep his paper money in order of denomination. $$\begin{array}{lcc} \hline \text { Age } & \text { Men } & \text { Keep Paper Money in Order, \% } \\\ \hline 21-34 & 25 & 9 \\ \hline 35-44 & 30 & 61 \\ \hline 45-54 & 15 & 80 \\ \hline 55 \text { and over } & 10 & 80 \\ \hline \end{array}$$ A man's wallet was retrieved and the paper money in it was kept in order of denomination. What is the probability that the wallet belonged to a male guest between the ages of 35 and 44 ?

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was black given that the second ball drawn was black?

In "The Numbers Game," a state lottery, four numbers are drawn with replacement from an urn containing balls numbered \(0-9\), inclusive. Find the probability that a ticket holder has the indicated winning ticket. One digit (the first, second, third, or fourth digit)
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