Answer:
Step-by-step explanation:
To find the steady matrix for this situation, we need to set up a system of equations based on the given probabilities.
Let's assume we have two groups of customers: Group A consists of customers who turn in one or more tickets, and Group B consists of customers who do not turn in tickets. We'll represent the steady matrix as follows:
| A_next | | p(A_next|A) p(A_next|B) | | A |
| B_next | = | p(B_next|A) p(B_next|B) | x | B |
Here, A_next and B_next represent the proportions of customers in each group for the next week, and A and B represent the proportions of customers in each group for the current week.
We are given the following probabilities:
During any week, 40% of the customers who turn in one or more tickets do not bother to turn in tickets the following week. This implies that the probability of customers in Group A remaining in Group A is 60% (since 100% - 40% = 60%). So, p(A_next|A) = 0.6.
On the other hand, 30% of the customers who do not turn in tickets will turn in one or more tickets the following week. This implies that the probability of customers in Group B moving to Group A is 30%. So, p(A_next|B) = 0.3.
Since the remaining customers who do not move from Group B will still be in Group B, the probability of customers in Group B remaining in Group B is 70% (since 100% - 30% = 70%). So, p(B_next|B) = 0.7.
To find p(B_next|A), we can use the fact that the total probability for each group's movement must sum to 1. Therefore:
p(A_next|A) + p(B_next|A) = 1
0.6 + p(B_next|A) = 1
p(B_next|A) = 0.4
Now we can construct the steady matrix:
| A_next | | 0.6 0.3 | | A |
| B_next | = | 0.4 0.7 | x | B |
This steady matrix represents the proportions of customers in each group for the next week based on the proportions for the current week. To interpret this matrix, we can assume an initial distribution of customers between the two groups and repeatedly multiply the matrix by itself to see how the proportions change over time. The steady matrix represents the stable proportions that the system converges to after multiple iterations, assuming no external factors affect the probabilities.
For example, if we start with an initial distribution of 80% of customers in Group A and 20% in Group B, we can calculate the steady proportions as follows:
| A_next | | 0.6 0.3 |^n | A |
| B_next | = | 0.4 0.7 | x | B |
where n represents the number of weeks. As n approaches infinity, the resulting proportions will converge to the steady proportions.