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Chapter 4: Problem 22

Suppose that we have a sample space with five equally likely experimental outcomes: \(E_{1}\). \\[ \begin{aligned} E_{2}, E_{3}, E_{4}, E_{5} . \text { Let } \\ \qquad \begin{aligned} A &=\left\\{E_{1}, E_{2}\right\\} \\ B &=\left\\{E_{3}, E_{4}\right\\} \\ C &=\left\\{E_{2}, E_{3}, E_{5}\right\\} \end{aligned} \end{aligned} \\] a. \(\quad\) Find \(P(A), P(B),\) and \(P(C)\). b. Find \(P(A \cup B)\). Are \(A\) and \(B\) mutually exclusive? c. \(\quad\) Find \(A^{c}, C^{c}, P\left(A^{c}\right),\) and \(P\left(C^{c}\right)\). d. Find \(A \cup B^{c}\) and \(P\left(A \cup B^{c}\right)\). e. Find \(P(B \cup C)\).

Short Answer

Expert verified
P(A) = 2/5, P(B) = 2/5, P(C) = 3/5; A and B are mutually exclusive, so P(A 鈭� B) = 4/5; P(A^c) = 3/5, P(C^c) = 2/5; P(A 鈭� B^c) = 3/5; P(B 鈭� C) = 4/5.

Step by step solution

01

Find Probability of Event A

Since the sample space has five equally likely outcomes, each outcome has a probability of \( \frac{1}{5} \). Event \( A \) consists of \( \{E_1, E_2\} \), so the probability \( P(A) \) is calculated as:\[P(A) = P(E_1) + P(E_2) = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}\]
02

Find Probability of Event B

Event \( B \) consists of \( \{E_3, E_4\} \). Similarly, the probability \( P(B) \) is:\[P(B) = P(E_3) + P(E_4) = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}\]
03

Find Probability of Event C

Event \( C \) includes \( \{E_2, E_3, E_5\} \). Thus, \( P(C) \) is calculated as:\[P(C) = P(E_2) + P(E_3) + P(E_5) = \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{3}{5}\]
04

Determine Mutual Exclusivity of A and B

Events \( A \) and \( B \) are mutually exclusive if they have no outcomes in common. Since \( A = \{E_1, E_2\} \) and \( B = \{E_3, E_4\} \) share no common outcomes, they are mutually exclusive.
05

Find Probability of A Union B

Since \( A \) and \( B \) are mutually exclusive, \( P(A \cup B) = P(A) + P(B) \):\[P(A \cup B) = \frac{2}{5} + \frac{2}{5} = \frac{4}{5}\]
06

Find Complement of A and Calculate P(A^c)

The complement of \( A \), denoted \( A^c \), contains all outcomes not in \( A \): \( A^c = \{E_3, E_4, E_5\} \). So, \( P(A^c) \) is:\[P(A^c) = 1 - P(A) = 1 - \frac{2}{5} = \frac{3}{5}\]
07

Find Complement of C and Calculate P(C^c)

The complement of \( C \), denoted \( C^c \), has outcomes not in \( C \): \( C^c = \{E_1, E_4\} \). Hence, \( P(C^c) \) is:\[P(C^c) = 1 - P(C) = 1 - \frac{3}{5} = \frac{2}{5}\]
08

Find A Union B Complement and Its Probability

First, find \( B^c \), the complement of \( B \): \( B^c = \{E_1, E_2, E_5\} \). Then, \( A \cup B^c \) includes \( \{E_1, E_2, E_5\} \). Thus,\[P(A \cup B^c) = \frac{3}{5}\]
09

Find Probability of B Union C

\( B \cup C = \{E_2, E_3, E_4, E_5\} \) represents outcomes in either \( B \) or \( C \), or both. Therefore,\[P(B \cup C) = \frac{4}{5}\]

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Key Concepts

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

Mutually Exclusive Events
In probability theory, mutually exclusive events refer to events that cannot happen simultaneously. This means if one event occurs, the other cannot. For example, if you toss a six-sided die, the result can be either a 3 or a 4, but not both at the same time. In the context of our exercise, events \( A \) and \( B \) are defined as \( A = \{E_1, E_2 \} \) and \( B = \{E_3, E_4 \} \). Since there are no common outcomes in \( A \) and \( B \), they are called mutually exclusive.

馃専 **Key Takeaways:**
* Mutually exclusive events have no overlap in their outcomes.
* The probability of both events occurring simultaneously is zero.
* For such events, the probability of either event occurring is the sum of their individual probabilities: \( P(A \cup B) = P(A) + P(B) \).
Sample Space
The sample space in probability theory is the set of all possible outcomes of a random experiment. Think of it as the complete "inventory" of potential results when an experiment is conducted. For the given example with five equally likely outcomes \( E_1, E_2, E_3, E_4, \) and \( E_5 \), the sample space \( S \) is \( \{E_1, E_2, E_3, E_4, E_5 \} \). Each outcome in the sample space is equally likely, with a probability calculated as \( \frac{1}{5} \), since they are equally likely.

馃 **Understanding Sample Space:**
* A comprehensive list of all possible individual outcomes of an experiment.
* Fundamental for calculating probabilities of events.
* Helps in identifying complements and unions of events.
Complementary Events
Complementary events are pairs of events where one event's occurrence means the other cannot occur, and vice versa. For any event \( A \) within a given sample space, the complement of \( A \) (denoted as \( A^c \)) includes all outcomes that are not part of \( A \).
In our exercise, the event \( A \) consists of \( \{E_1, E_2\} \), so its complement \( A^c \) is \( \{E_3, E_4, E_5\} \), as it includes all outcomes not in \( A \). Furthermore, the probability of the complement, \( P(A^c) \), is found by subtracting the probability of \( A \) from 1, resulting in \( P(A^c) = 1 - P(A) \).

馃幆 **Essential Points About Complementary Events:**
* The sum of the probabilities of an event and its complement is always 1.
* Complementary events cover all possible scenarios together in a sample space.
* Finding the probability of an event's complement can often be simpler than finding the event's probability directly.

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Most popular questions from this chapter

A company studied the number of lost-time accidents occurring at its Brownsville, Texas, plant. Historical records show that \(6 \%\) of the employees suffered lost-time accidents last year. Management believes that a special safety program will reduce such accidents to \(5 \%\) during the current year. In addition, it estimates that \(15 \%\) of employees who had lost-time accidents last year will experience a lost-time accident during the current year. a. What percentage of the employees will experience lost-time accidents in both years? b. What percentage of the employees will suffer at least one lost-time accident over the two-year period?

High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted \(18 \%\) of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was \(2375 .\) Let \(E, R,\) and \(D\) represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate \(P(E), P(R),\) and \(P(D)\). b. Are events \(E\) and \(D\) mutually exclusive? Find \(P(E \cap D)\). c. For the 2375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission? d. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Information about mutual funds provided by Morningstar Investment Research includes the type of mutual fund (Domestic Equity, International Equity, or Fixed Income) and the Morningstar rating for the fund. The rating is expressed from 1-star (lowest rating) to 5-star (highest rating). A sample of 25 mutual funds was selected from Morningstar Funds \(500(2008) .\) The following counts were obtained: \(\bullet\)Assume that one of these 25 mutual funds will be randomly selected in order to learn more about the mutual fund and its investment strategy. a. What is the probability of selecting a Domestic Equity fund? b. What is the probability of selecting a fund with a 4 -star or 5 -star rating? c. What is the probability of selecting a fund that is both a Domestic Equity fund and a fund with a 4 -star or 5 -star rating? d. What is the probability of selecting a fund that is a Domestic Equity fund or a fund with a 4 -star or 5 -star rating?Sixteen mutual funds were Domestic Equity funds. \(\bullet\)Thirteen mutual funds were rated 3 -star or less. \(\bullet\)Seven of the Domestic Equity funds were rated 4-star. \(\bullet\)Two of the Domestic Equity funds were rated 5 -star. Assume that one of these 25 mutual funds will be randomly selected in order to learn more about the mutual fund and its investment strategy. a. What is the probability of selecting a Domestic Equity fund? b. What is the probability of selecting a fund with a 4 -star or 5 -star rating? c. What is the probability of selecting a fund that is both a Domestic Equity fund and a fund with a 4 -star or 5 -star rating? d. What is the probability of selecting a fund that is a Domestic Equity fund or a fund with a 4 -star or 5 -star rating?

In the city of Milford, applications for zoning changes go through a two-step process: a review by the planning commission and a final decision by the city council. At step 1 the planning commission reviews the zoning change request and makes a positive or negative recommendation concerning the change. At step 2 the city council reviews the planning commission's recommendation and then votes to approve or to disapprove the zoning change. Suppose the developer of an apartment complex submits an application for a zoning change. Consider the application process as an experiment. a. How many sample points are there for this experiment? List the sample points. b. Construct a tree diagram for the experiment.

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