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Question

# In an office complex of 1000 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 85% chance that she will be at work tomorrow, and if the employee is absent today, there is a 60% chance that she will be absent tomorrow. Suppose that today there are 760 employees at work. Predict the number that will be at work tomorrow, the following day, and the day after that.

Solution

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Let $A$ be the following matrix:

$A=\begin{bmatrix} 0.85&0.40\\ 0.15&0.60 \end{bmatrix}$

Let $x$ be the vector containing the initial number of people at work and people who are absent:

$x=\begin{bmatrix} 760\\240 \end{bmatrix}$

$\text{\textcolor{#4257b2}{Number of people that will be at work tomorrow:}}$

$Ax=\begin{bmatrix} 0.85&0.40\\ 0.15&0.60 \end{bmatrix}\cdot\begin{bmatrix} 760\\240 \end{bmatrix}=\begin{bmatrix} 742\\258 \end{bmatrix}$

$\text{\textcolor{#4257b2}{Number of people that will be at work two days from now:}}$

$A^2x=\begin{bmatrix} 0.85&0.40\\ 0.15&0.60 \end{bmatrix}\cdot\begin{bmatrix} 742\\258 \end{bmatrix}\approx\begin{bmatrix} 734\\266 \end{bmatrix}$

$\text{\textcolor{#4257b2}{Number of people that will be at work three days from now:}}$

$A^3x=\begin{bmatrix} 0.85&0.40\\ 0.15&0.60 \end{bmatrix}\cdot\begin{bmatrix} 734\\266 \end{bmatrix}\approx\begin{bmatrix} 730\\270 \end{bmatrix}$

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