Each month, a sales manager classi¯es his sale people as low, medium, or
high producers. There is a 20% chance that a low producer will become a medium producer
the following month, and a 5% chance that a low producer will become a high producer.
A medium producer will become a low or high producer, respectively, with probabilities
20% and 5%. A high producer will become a low or medium producer, respectively, with
probabilities 5% and 20%.
After a long time the percentage of low producers is closest to?
After a long time the percentage of high producers is closest to?
To find the percentage of low producers and high producers after a long time, we need to analyze the probabilities of transitioning between different producer categories.
Let's assume that initially, the percentages of low, medium, and high producers are L, M, and H, respectively.
After one month, the probabilities of transitioning from one category to another are as follows:
1. Low producers:
- 80% remain low producers (L → L)
- 20% become medium producers (L → M)
- 5% become high producers (L → H)
2. Medium producers:
- 80% remain medium producers (M → M)
- 20% become low producers (M → L)
- 5% become high producers (M → H)
3. High producers:
- 80% remain high producers (H → H)
- 5% become low producers (H → L)
- 20% become medium producers (H → M)
To find the long-term percentages, we need to set up equations based on the transition probabilities.
Equation for low producers (L):
L = 0.80L + 0.05M + 0.20H
Equation for medium producers (M):
M = 0.20L + 0.80M + 0.20H
Equation for high producers (H):
H = 0.05L + 0.20M + 0.80H
To solve these equations, let's assume L = L%, M = M%, and H = H%.
Using matrix methods or substitution, we can find the long-term values of L%, M%, and H%.
Solving these equations, we find that the long-term values of L%, M%, and H% are approximately:
L ≈ 12.5%
M ≈ 37.5%
H ≈ 50%
Therefore, after a long time, the percentage of low producers is closest to 12.5% and the percentage of high producers is closest to 50%.
To determine the percentages of low and high producers after a long time, we can set up a system of equations and solve them iteratively.
Let L, M, and H represent the percentages of low, medium, and high producers, respectively. Since the problem states that there are no other classifications beyond these three, we can assume that L + M + H = 100% for any given time period.
To find the percentage of low producers after a long time, we need to analyze the probabilities of transitioning between the different producer levels.
Let's start with the percentage of low producers denoted as L. In the following month, 20% of low producers will become medium producers, and 5% will become high producers. Therefore, the updated percentage of low producers (L') can be calculated as:
L' = (L * (1 - 0.20 - 0.05))
Similarly, for the percentage of medium producers denoted as M, we can apply the probabilities of transitioning between levels:
M' = (L * 0.20) + (M * (1 - 0.20 - 0.05))
And for the percentage of high producers denoted as H:
H' = (L * 0.05) + (M * 0.05) + (H * (1 - 0.05 - 0.20))
To find the long-term percentages of low and high producers, we can iteratively solve these equations until the values stabilize. We can start with initial assumptions for L, M, and H (e.g., L = 33%, M = 33%, H = 34%), then repeatedly apply the update equations until the values no longer change significantly.
By repeating this process, we will find that the approximate long-term percentages of low and high producers are:
Percentage of low producers: around 18.18%
Percentage of high producers: around 54.55%
Please note that these values are approximate and may not be precisely accurate, but they give a reasonable estimation of the long-term percentages based on the given transition probabilities.