4. Suppose we have a fair coin with face 1 and 0. The game is to obtain in four consecutive flips either 1111 or 0011. The coin can be flipped as long as needed. Suppose that if 1111 appears first, John wins and if 0011 appears first, Tom win. a) Draw the transition of the game. [2 point] b) Find the winning probability for John and Tom. [4 point] c) Find the expected number of games to end. [3 point] d) If both of them start with $5, find the probability of John would maxed out Tom, and vice versa. [2 point] e) If John starts with $3 and Tom starts with $6, find the expected number of games until John maxed out Tom. [2 point] 5. Consider an irreducible Markov chain X on the state space S transition matrix {a, b, c, d, e, f} with P = 0 12230 250 0120 270 27 450 960 250 1500 170 47 0000157 0 ○ 16470 a) Draw the transition graph of the Markov chain. [1 point] b) Find the stationary distribution of the chain. [2 point] c) How would you check that this Markov chain is reversible? [3 point] d) Suppose that those chain determine the "up" and "down" of a process, with accept- able states A = {a, c, d}. Find the rate of breakdown. [3 point] e) Find the average time of the processes stay "up" and the processes stay "down". [2 point]
4. Suppose we have a fair coin with face 1 and 0. The game is to obtain in four consecutive flips either 1111 or 0011. The coin can be flipped as long as needed. Suppose that if 1111 appears first, John wins and if 0011 appears first, Tom win. a) Draw the transition of the game. [2 point] b) Find the winning probability for John and Tom. [4 point] c) Find the expected number of games to end. [3 point] d) If both of them start with $5, find the probability of John would maxed out Tom, and vice versa. [2 point] e) If John starts with $3 and Tom starts with $6, find the expected number of games until John maxed out Tom. [2 point] 5. Consider an irreducible Markov chain X on the state space S transition matrix {a, b, c, d, e, f} with P = 0 12230 250 0120 270 27 450 960 250 1500 170 47 0000157 0 ○ 16470 a) Draw the transition graph of the Markov chain. [1 point] b) Find the stationary distribution of the chain. [2 point] c) How would you check that this Markov chain is reversible? [3 point] d) Suppose that those chain determine the "up" and "down" of a process, with accept- able states A = {a, c, d}. Find the rate of breakdown. [3 point] e) Find the average time of the processes stay "up" and the processes stay "down". [2 point]
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.7: Probability
Problem 28E
Related questions
Question
Consider an irreducible Markov chain X on the state space S = {a, b, c, d, e, f} with transition matrix
a) Draw the transition graph of the Markov chain. [1 point]
b) Find the stationary distribution of the chain. [2 point]
c) How would you check that this Markov chain is reversible? [3 point]
d) Suppose that those chain determine the "up" and "down" of a process, with accept- able states A = {a,c,d}. Find the rate of breakdown. [3 point]
e) Find the average time of the processes stay "up" and the processes stay "down". [2 point]
![4. Suppose we have a fair coin with face 1 and 0. The game is to obtain in four consecutive
flips either 1111 or 0011. The coin can be flipped as long as needed. Suppose that if
1111 appears first, John wins and if 0011 appears first, Tom win.
a) Draw the transition of the game. [2 point]
b) Find the winning probability for John and Tom. [4 point]
c) Find the expected number of games to end. [3 point]
d) If both of them start with $5, find the probability of John would maxed out Tom,
and vice versa. [2 point]
e) If John starts with $3 and Tom starts with $6, find the expected number of games
until John maxed out Tom. [2 point]
5. Consider an irreducible Markov chain X on the state space S
transition matrix
{a, b, c, d, e, f} with
P =
0
12230 250
0120 270 27
450 960 250
1500 170 47
0000157
0
○ 16470
a) Draw the transition graph of the Markov chain. [1 point]
b) Find the stationary distribution of the chain. [2 point]
c) How would you check that this Markov chain is reversible? [3 point]
d) Suppose that those chain determine the "up" and "down" of a process, with accept-
able states A = {a, c, d}. Find the rate of breakdown. [3 point]
e) Find the average time of the processes stay "up" and the processes stay "down". [2
point]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b791892-bc02-471f-ae22-6feb2fc8a157%2F0f82fc39-d615-4ee4-b4a2-a127b3eb4dd5%2Fuassb0c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Suppose we have a fair coin with face 1 and 0. The game is to obtain in four consecutive
flips either 1111 or 0011. The coin can be flipped as long as needed. Suppose that if
1111 appears first, John wins and if 0011 appears first, Tom win.
a) Draw the transition of the game. [2 point]
b) Find the winning probability for John and Tom. [4 point]
c) Find the expected number of games to end. [3 point]
d) If both of them start with $5, find the probability of John would maxed out Tom,
and vice versa. [2 point]
e) If John starts with $3 and Tom starts with $6, find the expected number of games
until John maxed out Tom. [2 point]
5. Consider an irreducible Markov chain X on the state space S
transition matrix
{a, b, c, d, e, f} with
P =
0
12230 250
0120 270 27
450 960 250
1500 170 47
0000157
0
○ 16470
a) Draw the transition graph of the Markov chain. [1 point]
b) Find the stationary distribution of the chain. [2 point]
c) How would you check that this Markov chain is reversible? [3 point]
d) Suppose that those chain determine the "up" and "down" of a process, with accept-
able states A = {a, c, d}. Find the rate of breakdown. [3 point]
e) Find the average time of the processes stay "up" and the processes stay "down". [2
point]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill