A die is rolled successively until, for the first time, a number appears three consecutive times. By simulation, determine the approximate probability that it takes at least 50 rolls before we accomplish this event.
Read moreTable of Contents
1
Axioms of Probability
1.2
Sample Space and Events
1.4
Basic Theorems
2
Combinatorial Methods
2.2
Counting Principle
2.3
Permutations
2.4
Combinations
3
Conditional Probability and Independence
3.1
Conditional Probability
3.2
Law of Multiplication
3.3
Law of Total Probability
3.5
Independence
3.6
Applications of Probability to Genetics
4
Distribution Functions and Discrete Random Variables
4.2
Distribution Functions and Discrete Random Variables
4.4
Expectations of Discrete Random Variables
4.5
Variances and Moments of Discrete Random Variables
5
Special Discrete Distributions
5.1
Bernoulli and Binomial Random Variables
5.2
Poisson Random Variables
5.3
Other Discrete Random Variables
6
Continuous Random Variables
6.1
Probability Density Functions
6.2
Density Function of a Function of a Random Variable
6.3
Expectations and Variances
7
Special Continuous Distributions
7.2
Normal Random Variables
7.3
Exponential Random Variables
7.5
Beta Distributions
7.6
Survival Analysis and Hazard Functions
8
Bivariate Distributions
8.1
Joint Distributions of Two Random Variables
8.2
Independent Random Variables
8.3
Conditional Distributions
8.4
Transformations of Two Random Variables
9
Multivariate Distributions
9.1
Joint Distributions of n > 2 Random Variables
9.2
Order Statistics
9.3
Order Statistics
10
More Expectations and Variances
10.1
Expected Values of Sums of Random Variables
10.2
Covariance
10.3
Correlation
10.4
Conditioning on Random Variables
10.5
Bivariate Normal Distribution
11
Sums of Independent Random Variables and Limit Theorems
11.1
Moment-Generating Functions
11.2
Sums of Independent Random Variables
11.3
Markov and Chebyshev Inequalities
11.4
Laws of Large Numbers
12
Stochastic Processes
12.2
More on Poisson Processes
12.3
Markov Chains
12.4
Continuous-Time Markov Chains
12.5
Brownian Motion
13
Simulation
13.2
Simulation of Combinatorial Problems
13.4
Simulation of Random Variables
13.5
Monte Carlo Method
Textbook Solutions for Fundamentals of Probability, with Stochastic Processes
Chapter 13 Problem 1
Question
A die is rolled successively until, for the first time, a number appears three consecutive times. By simulation, determine the approximate probability that it takes at least 50 rolls before we accomplish this event.
Solution
The first step in solving 13 problem number 1 trying to solve the problem we have to refer to the textbook question: A die is rolled successively until, for the first time, a number appears three consecutive times. By simulation, determine the approximate probability that it takes at least 50 rolls before we accomplish this event.
From the textbook chapter Simulation you will find a few key concepts needed to solve this.
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full solution
Title
Fundamentals of Probability, with Stochastic Processes 3
Author
Saeed Ghahramani
ISBN
9780131453401
A die is rolled successively until, for the first time, a
Chapter 13 textbook questions
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Chapter 13: Problem 1 Fundamentals of Probability, with Stochastic Processes 3
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Chapter 13: Problem 2 Fundamentals of Probability, with Stochastic Processes 3
In an election, the Democratic candidate obtained 3586 votes and the Republican candidate obtained 2958. Use simulations to find the approximate probability that the Democratic candidate was ahead during the entire process of counting the votes. Answer: Approximately 0.096.
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Chapter 13: Problem 3 Fundamentals of Probability, with Stochastic Processes 3
The probability that a bank refuses to finance an applicant is 0.35. Using simulation, find the approximate probability that of 300 applicants more than 100 are refused.
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Chapter 13: Problem 4 Fundamentals of Probability, with Stochastic Processes 3
There are five urns, each containing 10 white and 15 red balls. A ball is drawn at random from the first urn and put into the second one. Then a ball is drawn at random from the second urn and put into the third one and the process is continued. By simulation, determine the approximate probability that the last ball is red. Answer: 0.6.
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Chapter 13: Problem 5 Fundamentals of Probability, with Stochastic Processes 3
In a small town of 1000 inhabitants, someone gossips to a random person, who in turn tells the story to another random person, and so on. Using simulation, calculate the approximate probability that before the story is told 150 times, it is returned to the first person.
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Chapter 13: Problem 6 Fundamentals of Probability, with Stochastic Processes 3
A city has n taxis numbered 1 through n. A statistician takes taxis numbered 31, 50, and 112 on three random occasions. Based on this information, determine, using simulation, which of the numbers 112 through 120 is a better estimate for n. Hint: For each n (112 n 120), repeat the following experiment a large number of times: Choose three random numbers from {1, 2, . . . , n} and check to see if they are 31, 50, and 112.
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Chapter 13: Problem 7 Fundamentals of Probability, with Stochastic Processes 3
Suppose that an airplane passenger whose itinerary requires a change, of airplanes in Ankara, Turkey, has a 4% chance, independently, of losing each piece of his or her luggage. Suppose that the probability of losing each piece of luggage in this way is 5% at DaVinci airport in Rome, 5% at Kennedy airport in NewYork, and 4% at OHare airport in Chicago. Dr. May travels from Bombay to San Francisco with three suitcases. He changes airplanes in Ankara, Rome, New York, and Chicago. Using simulation, find the approximate probability that one of Dr. Mays suitcases does not reach his destination with him.
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