are: When it is Coin's turn, he or she flips the coin. If the coin turns up heads, Coin wins the game. If the coin turns up tails, it is Die's turn. When it is Die's turn, he or she rolls the die. If the die turns up 1, Die wins. If the die turns up 6, it is Coin's turn. Otherwise, Die rolls again. When it is Coin's turn, the probability is 1/2 that Coin will win and 1/2 that it will become Die's turn. When it is Die's turn, the probabilities are 1/6 that Die will roll a 1 and win 1/6 that Die will roll a 6 and it will become Coin's turn, and 2/3 that Die will roll a 2, 3, 4, or 5 and have another turn. To describe this process as a Markov chain, we define four states of the process: State 0: Coin has won the game. State 1: It is Coin's turn. State 2: Die has won the game. State 3: It is Die's turn. - (a) Give the transition matrix P for this Markov chain. (Hint: I have done the first two rows of this matrix for you P = 1 0 0 0 1/2 0 0 1/2 (b) Use the fundamental matrix method, identify what the matrices Q and R are. Find (I-Q)-¹ and (I - Q)-¹R. (c) Use the result from part (b) to answer the following questions: What is the probability that Coin will win if Coin goes first? What is the probability that Coin will win if Die goes first? If Coin goes first, what is the expected number of turns for Coin before the game ends? And what is the expected number of total turns of this game before it ends? If Die goes first, what is the expected number of turns for Die before the game ends? And what is the expected number of total turns of this game before it ends?
are: When it is Coin's turn, he or she flips the coin. If the coin turns up heads, Coin wins the game. If the coin turns up tails, it is Die's turn. When it is Die's turn, he or she rolls the die. If the die turns up 1, Die wins. If the die turns up 6, it is Coin's turn. Otherwise, Die rolls again. When it is Coin's turn, the probability is 1/2 that Coin will win and 1/2 that it will become Die's turn. When it is Die's turn, the probabilities are 1/6 that Die will roll a 1 and win 1/6 that Die will roll a 6 and it will become Coin's turn, and 2/3 that Die will roll a 2, 3, 4, or 5 and have another turn. To describe this process as a Markov chain, we define four states of the process: State 0: Coin has won the game. State 1: It is Coin's turn. State 2: Die has won the game. State 3: It is Die's turn. - (a) Give the transition matrix P for this Markov chain. (Hint: I have done the first two rows of this matrix for you P = 1 0 0 0 1/2 0 0 1/2 (b) Use the fundamental matrix method, identify what the matrices Q and R are. Find (I-Q)-¹ and (I - Q)-¹R. (c) Use the result from part (b) to answer the following questions: What is the probability that Coin will win if Coin goes first? What is the probability that Coin will win if Die goes first? If Coin goes first, what is the expected number of turns for Coin before the game ends? And what is the expected number of total turns of this game before it ends? If Die goes first, what is the expected number of turns for Die before the game ends? And what is the expected number of total turns of this game before it ends?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:10. There are two players, Coin and Die. Coin has a coin, and Die has a single six-sided die. The rules
are:
When it is Coin's turn, he or she flips the coin. If the coin turns up heads, Coin wins the
game. If the coin turns up tails, it is Die's turn.
When it is Die's turn, he or she rolls the die. If the die turns up 1, Die wins. If the die turns up
6, it is Coin's turn. Otherwise, Die rolls again.
When it is Coin's turn, the probability is 1/2 that Coin will win and 1/2 that it will become Die's
turn. When it is Die's turn, the probabilities are
1/6 that Die will roll a 1 and win
1/6 that Die will roll a 6 and it will become Coin's turn, and
2/3 that Die will roll a 2, 3, 4, or 5 and have another turn.
To describe this process as a Markov chain, we define four states of the process:
State 0 : Coin has won the game.
State 1: It is Coin's turn.
State 2: Die has won the game.
State 3: It is Die's turn.
(a) Give the transition matrix P for this Markov chain.
(Hint: I have done the first two rows of this matrix for you P =
1
0 0
1/2 0 0
0
1/2
(b) Use the fundamental matrix method, identify what the matrices Q and R are. Find (I — Q)−¹
and (I — Q)−¹R.
-
(c) Use the result from part (b) to answer the following questions:
What is the probability that Coin will win if Coin goes first? What is the probability
that Coin will win if Die goes first?
If Coin goes first, what is the expected number of turns for Coin before the game
ends? And what is the expected number of total turns of this game before it ends?
If Die goes first, what is the expected number of turns for Die before the game ends?
And what is the expected number of total turns of this game before it ends?
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