## Modelling Categories of Disease and Sufferer With Matrices

Suppose there are three age categories of individuals. Each individual has the same disease, to a different degree. The probability of individuals in each category having the disease to each degree is given in the table below.
 Age\Severity Slight Mild Severe 0 - 30 0.5 0.3 0.2 30 - 60 0.4 0.3 0.3 60 - 90 0.1 0.4 0.5
One thousand people are selected. 300 are age 0 - 30, 500 are aged 30 - 60, and 200 are aged 60 - 90. How many people would you expect in each disease category?
We can represent the table by the mat ix
$\left( \begin{array}{ccc} 0.5 & 0.3 & 0.2 \\ 0.4 & 0.3 & 0.3 \\ 0.1 & 0.4 & 0.5 \end{array} \right)$
, the the people above by the vector
$\begin{pmatrix}300\\500\\200\end{pmatrix}$
.
$\left( \begin{array}{ccc} 0.5 & 0.3 & 0.2 \\ 0.4 & 0.3 & 0.3 \\ 0.1 & 0.4 & 0.5 \end{array} \right)\begin{pmatrix}300\\500\\200\end{pmatrix}= \begin{pmatrix}370\\320\\310\end{pmatrix}$

370 people have a slight form of the disease, 320 have a mild form of the disease, and 310 have a severe form of the disease.