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 the situation as a Markov chain, let's define the states as follows:
State 0: Represents zero consecutive games won by Doug
State 1: Represents one consecutive game won by Doug
State 2: Represents two consecutive games won by Doug
State 3: Represents three or more consecutive games won by Doug
Now, let's determine the transition probabilities for each state. We are given the following information:
- If Doug wins a game, he will spot Bob five points in the next game.
- If Doug wins again, he will spot Bob ten points in the next game.
- If he still wins the next game, he will spot Bob fifteen points, and so on.
Using this information, we can determine the transition probabilities as follows:
- From State 0, the probability of transitioning to State 1 (Doug wins) is 0.9 (90% chance).
- From State 0, the probability of transitioning to State 0 (Bob wins) is 0.1 (10% chance).
- From State 1, the probability of transitioning to State 2 (Doug wins) is 0.9 (90% chance).
- From State 1, the probability of transitioning to State 0 (Bob wins) is 0.1 (10% chance).
- From State 2, the probability of transitioning to State 3 (Doug wins) is 0.9 (90% chance).
- From State 2, the probability of transitioning to State 0 (Bob wins) is 0.1 (10% chance).
- From State 3, the probability of transitioning to State 3 (Doug wins) is 1.0 (100% chance).
Now, let's write the transition matrix:
State 0 State 1 State 2 State 3
State 0 0.1 0.9 0 0
State 1 0.1 0 0.9 0
State 2 0.1 0 0 0.9
State 3 0 0 0 1.0
Next, let's find the steady-state vector for the system. This vector represents the long-term proportions of games in each state. We need to find a vector 𝐯 that satisfies the equation 𝐯𝑇𝑀 = 𝐯𝑇, where 𝑀 is the transition matrix.
To find the steady-state vector, we solve the equation (𝑀−𝐼)𝐯 = 0, where 𝐼 is the identity matrix. In this case, we have:
[ -0.9 0.9 0 0 ]
[ 0.1 -1.0 0 0 ]
[ 0.1 0 -1.0 0 ]
[ 0 0 0 -1.0 ]
To solve this system, we row-reduce the matrix to echelon form:
[ 1 -1 0 0 ]
[ 0 1 -1 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
From the row-reduced echelon form, we can see that the system has infinitely many solutions since we have a row of zeros. One possible steady-state vector is:
𝐯 = [1, 1, 1, 1]
Finally, let's determine the proportion of games that Doug will win in the long run. To do this, we can take the steady-state vector and sum up the probabilities for the states where Doug wins:
Proportion of games Doug will win = 𝐯0 + 𝐯1 + 𝐯2
= 1 + 1 + 1
= 3
Therefore, in the long run, Doug will win 3/4 or 75% of the games.
To model this situation as a Markov chain, we will use the number of consecutive games won by Doug as the states. There will be four states representing zero, one, two, and three or more consecutive games won by Doug.
Let's label these states as follows:
State 0: Zero consecutive games won by Doug
State 1: One consecutive game won by Doug
State 2: Two consecutive games won by Doug
State 3: Three or more consecutive games won by Doug
The transition matrix will represent the probabilities of transitioning between these states. We are given the probabilities of Bob winning the next game based on the points spotted to him.
From the given information:
- If Doug wins a game, he will spot Bob five points in the next game, and Bob wins 30% of the time.
- If Doug wins a game again, he will spot Bob ten points in the next game, and Bob wins 50% of the time.
- If Doug wins a game for the third consecutive time, he will spot Bob fifteen points in the next game, and Bob wins 60% of the time.
Using this information, we can construct the transition matrix:
```
| P(0 -> 0) P(0 -> 1) P(0 -> 2) P(0 -> 3) |
M = | P(1 -> 0) P(1 -> 1) P(1 -> 2) P(1 -> 3) |
| P(2 -> 0) P(2 -> 1) P(2 -> 2) P(2 -> 3) |
| P(3 -> 0) P(3 -> 1) P(3 -> 2) P(3 -> 3) |
```
To find the values of the transition matrix, we need to use the given probabilities of Bob winning with the spotted points.
From the given information:
- With a five-point advantage, Bob wins 30% of the time.
- With a ten-point advantage, Bob wins 50% of the time.
- With a fifteen-point advantage, Bob wins 60% of the time.
Using these probabilities, we can fill the transition matrix as follows:
```
| 0.7 0.3 0 0 |
M = | 0.5 0 0.5 0 |
| 0 0.5 0 0.5 |
| 0 0 0.5 0.5 |
```
Now, we can find the steady-state vector for the system. The steady-state vector represents the long-term proportion of time spent in each state.
To find the steady-state vector, we solve the equation:
```
M * X = X
```
where `X` is the steady-state vector.
Solving this equation, we can find the steady-state vector `X`.
Finally, to determine the proportion of games that Doug will win in the long run, we look at the steady-state vector `X`. The proportion of games Doug wins will be the value corresponding to State 3 (three or more consecutive games won by Doug) in the steady-state vector `X`.
By finding the transition matrix, solving for the steady-state vector, and analyzing the proportion of games Doug wins in the long run, we can determine the desired information in this problem.