Final answer:
The stochastic matrix for the given Markov chain is [[0.81, 0.39], [0.19, 0.61]]. To predict the number of employees at work in five days, one would use this matrix raised to the fifth power, M^5, and apply it to the initial state vector. The steady-state vector is found by solving Mx = x where x sums to 1.
Step-by-step explanation:
To find the stochastic matrix for the Markov chain, we classify the states based on whether an employee is at work or absent. The matrix consists of two probabilities for each state: the probability of an employee being at work tomorrow given that they are at work today, and the probability of them being absent tomorrow given that they're absent today.
- The probability of an employee being at work tomorrow if they were at work today is 81%.
- The probability of an employee being at work tomorrow if they were absent today is 39% (100% - 61%).
- The probability of an employee being absent tomorrow if they were at work today is 19% (100% - 81%).
- The probability of an employee being absent tomorrow if they were absent today is 61%.
Thus, the stochastic matrix M is as follows:
M = [
[0.81, 0.39],
[0.19, 0.61]
]
To predict how many employees will be at work five days from now, starting with 882 employees at work today, we use the stochastic matrix raised to the fifth power (M^5) and apply it to the initial state vector (the number of employees at work and absent today).
The steady-state vector is found by solving the equation Mx = x for the vector x, which satisfies the condition that the sum of its components is 1. This vector indicates the long-term proportion of employees at work and absent.