Consider the following situation:
There are 75 students in Gr. 11 Applied
Math this year, there are 90 students in
Gr. 11 Pre-Calculus
It is predicted that 85% of the Applied
students will continue and the rest will
switch to Pre-Calculus
Also, 70% of the Precalculus students will
stay in PC and the rest will switch to
Applied.
Q: How many students will be in each
program after the change?
To calculate the number of students in each program
we consider the students STAYING and the students
SWITCHING into
APP: (75)(0.85) + (90)(0.30) = 91 students
85% of 75 students STAY
If 70% stay in PC, 30%
switch to Applied
PC: (90)(0.70) + (75)(0.15) = 74 students
70% of 90 students STAY
If 85% stay in Applied,
15% switch to PC
We completed a manual calculation
of a situation that can be
represented with matrices
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A transition is a change from one group
to another. These can be represented
by Transition Diagrams
A transition diagram consists of:
- Nodes: represent the different groups
- Arcs: represent the change (movement)
APPLIED
PC
NOTE: The number on the arc represents the percentage
moving along that arc (we use percentages instead of
numbers because we can re-use the transition pattern with a
new set of numbers)
NOTE: All arrows leaving a node need to add up to 100%
(we need to account for ALL movement)
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Once we have a transition diagram, we
create a Transition Matrix
Transition matrices are:
-Always square matrices
-One row for each node
-One column for each node (in the same
order)
APPLIED
PC
-The elements are the changes (%)
FROM row TO column
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Next, we need to create an Initial
Population matrix to represent the
starting numbers of each group
The initial population matrix is:
-always a row matrix
-one column for each node (same order
as transition matrix)
-elements contain the starting numbers
of each node
To find the new population AFTER the
change:
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Example:
Studies show that each year,
5% of people living in Cities move to the
Country
15% of people living in the Country move
to the City
If 1500 people live in the City and 850
live in the country in 2011, how many
people live in each place in 2012?
If the change continues, how many
people will live in each place in 2013?
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