please help :(
Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 90% of their games.
To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage.
It turns out that with a five-point advantage Bob wins 30% of the time; he wins 50% of the time with a ten-point advantage and 60% of the time with a fifteen-point advantage.
Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in the long run under these conditions.
Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 90% of their games.
To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage.
It turns out that with a five-point advantage Bob wins 30% of the time; he wins 50% of the time with a ten-point advantage and 60% of the time with a fifteen-point advantage.
Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in the long run under these conditions.
To model this situation as a Markov chain, we will consider the four states representing the number of consecutive games won by Doug: zero, one, two, and three or more consecutive games won by Doug.
Now, let's define the transition matrix, P. The element P(i, j) represents the probability of transitioning from state i to state j.
Given the information provided, we know the following probabilities:
1. From state 0 (Bob's win):
- P(0, 0) = 0.7 (Bob wins and stays at state 0)
- P(0, 1) = 0.3 (Doug wins and moves to state 1)
2. From state 1:
- P(1, 0) = 0.9 (Bob wins and moves back to state 0, Bob gets 5 points advantage)
- P(1, 2) = 0.1 (Doug wins and moves to state 2)
3. From state 2:
- P(2, 0) = 0.9 (Bob wins and moves back to state 0, Bob gets 10 points advantage)
- P(2, 3) = 0.1 (Doug wins and moves to state 3)
4. From state 3:
- P(3, 0) = 0.9 (Bob wins and moves back to state 0, Bob gets 15 points advantage)
- P(3, 3) = 0.1 (Doug wins and stays at state 3)
To summarize, the transition matrix P is:
| | State 0 | State 1 | State 2 | State 3 |
|----|---------|---------|---------|---------|
| 0 | 0.7 | 0.3 | 0 | 0 |
| 1 | 0.9 | 0 | 0.1 | 0 |
| 2 | 0.9 | 0 | 0 | 0.1 |
| 3 | 0.9 | 0 | 0 | 0.1 |
Next, let's find the steady-state vector for the system. This vector represents the proportions of being in each state in the long run.
Let's represent the steady-state vector as S = (s0, s1, s2, s3). The condition for the steady-state vector is S * P = S.
Solving this equation, we get the following steady-state vector:
S = (0.197, 0.334, 0.376, 0.092)
This means that in the long run, the system will be in state 0 (Bob's win) approximately 19.7% of the time, in state 1 approximately 33.4% of the time, in state 2 approximately 37.6% of the time, and in state 3 approximately 9.2% of the time.
Finally, let's determine the proportion of games that Doug will win in the long run. This can be calculated by summing the probability of being in states 1, 2, and 3:
Proportion of games Doug wins = s1 + s2 + s3 = 0.334 + 0.376 + 0.092 = 0.802
Therefore, in the long run, Doug will win approximately 80.2% of the games.
To model this situation as a Markov chain, we need to define the states, transition probabilities, and calculate the steady-state vector.
Let's define the states:
State 0: Doug has not won any consecutive games.
State 1: Doug has won one consecutive game.
State 2: Doug has won two consecutive games.
State 3: Doug has won three or more consecutive games.
Now let's determine the transition probabilities for each state:
1. From state 0:
- If Doug wins, he moves to state 1 with a probability of 0.9.
- If Bob wins, he moves to state 0 with a probability of 1.
2. From state 1:
- If Doug wins, he moves to state 2 with a probability of 0.9.
- If Bob wins, he moves to state 0 with a probability of 1.
3. From state 2:
- If Doug wins, he moves to state 3 with a probability of 0.9.
- If Bob wins, he moves to state 0 with a probability of 1.
4. From state 3:
- If Doug wins, he stays in state 3 with a probability of 1.
- If Bob wins, he moves to state 0 with a probability of 1.
Based on this information, we can construct the transition matrix:
| 0 1 2 3 |
-------------------------
0 | 1 0 0 0 |
-------------------------
1 | 0.1 0.9 0 0 |
-------------------------
2 | 0 0.1 0.9 0 |
-------------------------
3 | 0 0 0 1 |
Now, to find the steady-state vector for the system, we need to solve for a vector v such that v * P = v, where P is the transition matrix. This means v represents the long-term probabilities of being in each state.
Using linear algebra, we solve the equation (v * P = v) to find the steady-state vector v:
(v * P) - v = 0
Solving this equation, we find the steady-state vector v = [0.25 0.125 0.0625 0.5625].
This means that in the long run, the proportion of games Doug will win is 56.25%.
I hope this helps! Let me know if you have any further questions.