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Chapter 7: Problem 20

Let \(S\) be any sample space and let E, \(\boldsymbol{F}\), and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(\cup, \cap\), and . The event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs

Short Answer

Expert verified
The event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs can be represented using the symbols \(\cup\), \(\cap\), and \(\complement\) as follows: \(E \cap \left(F^\complement \cap G^\complement\right)\)

Step by step solution

01

Identify the Symbols

Before we start, let's recall the meaning of these symbols: 1. \(\cup\): The union symbol denotes the event where either of the events (or both) occur. Mathematically, \(A \cup B\) represents the event where either \(A\) or \(B\) (or both) occurs. 2. \(\cap\): The intersection symbol denotes the event where both events occur simultaneously. Mathematically, \(A \cap B\) represents the event where both \(A\) and \(B\) occur. 3. \(\complement\): The complement symbol denotes the event where the event does NOT occur. Mathematically, \(A^\complement\) represents the event where \(A\) does not occur. We will use these symbols to describe the event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs.
02

Working with Events

Our goal is to describe the event that \(E\) occurs but neither of the events \(F\) or \(G\) occurs. First, let's break this down into two separate conditions: 1. \(E\) occurs: This is simply event \(E\). 2. Neither \(F\) nor \(G\) occurs: To describe this, we will use the complement symbol, denoting that both \(F\) and \(G\) do not occur. Mathematically, this is represented as \(F^\complement \cap G^\complement\) (i.e., both \(F\) and \(G\) do not occur).
03

Combining the Conditions

Now, we need to combine these conditions to describe the event we're looking for. Since we want \(E\) to occur AND neither \(F\) nor \(G\) to occur, we can represent this by taking the intersection of these two events: \(E \cap \left(F^\complement \cap G^\complement\right)\) And that's our answer! This expression represents the event where \(E\) occurs but neither of the events \(F\) or \(G\) occurs.

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Most popular questions from this chapter

Let \(E\) and \(F\) be two events of an experiment with sample space \(S\). Suppose \(P(E)=.6, P(F)=.4\), and \(P(E \cap F)=\) .2. Compute: a. \(P(E \cup F)\) b. \(P\left(E^{c}\right)\) c. \(P\left(F^{c}\right)\) d. \(P\left(E^{c} \cap F\right)\)

In a Los Angeles Times poll of 1936 California residents conducted in February 2004 , the following question was asked: Do you favor or oppose an amendment to the U.S. Constitution barring same-sex marriage? The following results were obtained: $$ \begin{array}{lccc} \hline \text { Opinion } & \text { Favor } & \text { Oppose } & \text { Don't know } \\ \hline \text { Respondents } & 910 & 891 & 135 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

The number of cars entering a tunnel leading to an airport in a major city over a period of 200 peak hours was observed, and the following data were obtained: $$ \begin{array}{rc} \hline \begin{array}{l} \text { Number of } \\ \text { Cars, } x \end{array} & \begin{array}{c} \text { Frequency of } \\ \text { Occurrence } \end{array} \\ \hline 01000 & 15 \\ \hline \end{array} $$ a. Describe an appropriate sample space for this experiment. b. Find the empirical probability distribution for this experiment.

In a survey of 2000 adults 18 yr and older conducted in 2007 , the following question was asked: Is your family income keeping pace with the cost of living? The results of the survey follow: $$ \begin{array}{lcccc} \hline & \begin{array}{c} \text { Falling } \\ \text { behind } \end{array} & \begin{array}{c} \text { Staying } \\ \text { even } \end{array} & \begin{array}{c} \text { Increasing } \\ \text { faster } \end{array} & \begin{array}{c} \text { Don't } \\ \text { know } \end{array} \\ \hline \text { Respondents } & 800 & 880 & 240 & 80 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\).
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