Book cover for A First Course in Probability

A First Course in Probability

Sheldon Ross

ISBN #9780136033134

8th Edition

502 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section lays the groundwork for probability theory by defining sample spaces, events, and the basic operations among events. It introduces the axioms of probability, ensuring that all probability assignments are consistent and logical. The text covers both the classical frequency approach and the subjective interpretation of probability, enriching our understanding of real-world problems. Additionally, the concepts of conditional probability and independence provide tools for updating probabilities in light of new information and for analyzing complex events.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Definition: Explores advanced probability models by extending fundamental theories to more complex systems. Topics include random processes (e.g., the Poisson process), Markov chains (with transition matrices, Chapman–Kolmogorov equations, and stationary distributions), and key ideas in information theory such as surprise, entropy, and coding (both noiseless and noisy).

Explores advanced probability models by extending fundamental theories to more complex systems. Topics include random processes (e.g., the Poisson process), Markov chains (with transition matrices, Chapman–Kolmogorov equations, and stationary distributions), and key ideas in information theory such as surprise, entropy, and coding (both noiseless and noisy). •

Example Problems

Example 1

A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

Example 2

In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let $E_{n}$ denote the event that $n$ rolls are necessary to complete the experiment. What points of the sample space are contained in $E_{n} ?$ What is $\left(\bigcup_{1}^{\infty} E_{n}\right)^{c} ?$

Example 3

Two dice are thrown. Let $E$ be the event that the sum of the dice is odd, let $F$ be the event that at least one of the dice lands on $1,$ and let $G$ be the event that the sum is $5 .$ Describe the events $E F, E \cup F, F G, E F^{c},$ and $E F G$.

Example 4

$A, B,$ and $C$ take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined by$$ S=\left\{\begin{array}{l}1,01,001,0001, \ldots, \\ 0000 \cdots\end{array}\right.$$ (a) Interpret the sample space. (b) Define the following events in terms of $S:$ (i) $A$ wins $=A$ (ii) $B$ wins $=B$ (iii) $(A \cup B)^{c}$ Assume that $A$ flips first, then $B$, then $C$, then $A,$ and so on.

Example 5

A system is comprised of 5 components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector $\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)$ where $x_{i}$ is equal to 1 if component $i$ is working and is equal to 0 if component $i$ is failed. (a) How many outcomes are in the sample space of this experiment? (b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components $1,3,$ and 5 are all working. Let $W$ be the event that the system will work. Specify all the outcomes in $W$ (c) Let $A$ be the event that components 4 and 5 are both failed. How many outcomes are contained in the event $A ?$ (d) Write out all the outcomes in the event $A W$.
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Step-by-Step Explanations

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Common Mistakes

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