Sheldon Ross
ISBN #9780136033134
8th Edition
502 Questions
Homework Questions
A First Course in Probability is a comprehensive introduction that lays out the theoretical foundations of probability and its practical applications. The book starts with combinatorial analysis—covering essential counting techniques like permutations, combinations, and advanced methods for indistinguishable objects—before introducing the axioms that underlie all probability theory. It then methodically develops the concepts of conditional probability, independence, and the behavior of random variables in both discrete and continuous settings, equipping readers with the tools to handle a wide range of probabilistic problems. Advanced topics, including joint distributions, simulation methods, limit theorems, and the connections between Markov chains, entropy, and coding theory, further enhance its value as a robust resource for anyone looking to master probability theory.
Chapter 1
Combinatorial Analysis
Chapter 2
Axioms of Probability
Chapter 3
Conditional Probability and Independence
Chapter 4
Random Variables
Chapter 5
Continuous Random Variables
Chapter 6
Jointly Distributed Random Variables
Chapter 7
Properties of Expectation
Chapter 8
Limit Theorems
Chapter 9
Additional Topics in Probability
Chapter 10
Simulation
Problem 1
Consider the grid of points shown here. Suppose that, starting at the point labeled $A,$ you can go one step up or one step to the right at each move. This procedure is continued until the point labeled $B$ is reached. How many different paths from $A$ to $B$ arc possible? Hint: Note that to reach $B$ from $A$, you must take 4 steps to the right and 3 steps upward.
Ahmad Reda Numerade Educator
Problem 2
A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?
Problem 3
An urn contains 3 red and 7 black balls. Players $A$ and $B$ withdraw balls from the urn consecutively until a red ball is selected. Find the probability that $A$ selects the red ball. $(A$ draws the first ball, then $B,$ and so on. There is no replacement of the balls drawn.)
Alexander Cheng Numerade Educator
Problem 4
In Problem $5,$ for $n=3,$ if the coin is assumed fair, what are the probabilities associated with the values that $X$ can take on?
Bryan Meares Numerade Educator
Problem 5
The probability of the closing of the $i$th relay in the circuits shown in Figure 3.4 is given by $p_{i}, i=1,2$ $3,4,5 .$ If all relays function independently, what is the probability that a current flows between $A$ and $B$ for the respective circuits? Hint for $(b):$ Condition on whether relay 3 closes.
Problem 6
Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p,$ then he or she will receive a score of $1-(1-p)^{2} \quad$ if it does rain $1-p^{2} \quad$ if it does not rain We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability $p^{*},$ what value of $p$ should he or she assert so as to maximize the expected score?
Harsh Gadhiya Numerade Educator
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