Introduction to solving problems with transition matrices.

1. Transition Matrices
jump.in.motion

2. An Example
A small community has a workforce of 1800 people, with 1680 employed
and 120 unemployed. During the course of one year, 10% of the employed
workers will lose their jobs and 60% of the unemployed will find jobs.
The quot;Initial State Matrixquot; The quot;Transition Diagramquot;
E U
S=
E U E U
S=
The quot;Transition Matrixquot;
E U
E
T= U
E U
T= E
U

3. Predicting the future ...
A small community has a workforce of 1800 people, with 1680 employed
and 120 unemployed. During the course of one year, 10% of the employed
workers will lose their jobs and 60% of the unemployed will find jobs.
After 1 Year
After 2 years
After 5 Years

5. After 50 years
Demonstrate stabilization ...

6. You Try ...
The Sudzmore Soap Company sells laundry detergent in two-litre and
five-litre packages. Their research shows that 34% of the people buying
the small package will switch to the large package for their next purchase,
and 12% of the buyers of the large package will switch to the small
package for their next purchase. The original market share was 55% for
the small package and 45% for the large package.
(a) Determine the market share for each size in the next round of
purchases.
small: 42%
large: 58%
(b) Does the market share for each size ever stabilize? If so, find
out when that occurs and what the market share for each size will
be. Stabilizes after 7 rounds of purchasing.
Market share is small: 26% and large: 74%

7. A Different Kind of Example ...
A small store in a remote community
sells three brands of soft drinks. The
three brands are Popsie, Sparkle, and
Fizz. The current market share is 60%
for Popsie, 31% for Sparkle, 9% for
Fizz. The buying trends for the three
brands over the past months is shown
on the diagram below. Since the store is
located in a remote community, the
store manager needs to place orders 12 months in advance. (The winter
roads to the community are operational for a few weeks of the year, and
airfare for soft drinks is too expensive.)
(a) Determine the anticipated monthly consumption 12 months from now.
Express your answers as percentages of the total consumption.
(b) The current consumption per month is 1600 cans and the store manager
expects consumption to increase 15% in 12 months. How many cans of each
brand of drink will likely be sold a year from now?

8. The current market share is 60% for Popsie, 31% for Sparkle, 9% for Fizz.
(b) The current consumption per month is 1600 cans and the store manager
expects consumption to increase 15% in 12 months. How many cans of each
brand of drink will likely be sold a year from now?

9. HOMEWORK
Suppose that for a “Winnipeg spring”, long run data suggests that there is a
28% chance that if today’s weather is good, then so will the next days’ be.
Conversely, if today is unpleasant, there is a 61% chance that the next day
will also be bad weather. Suppose further that these two are complementary
states. (i.e. Each day the weather is either nice or bad.) This information can
then be represented using a 2 × 2 transition matrix.
(a) Use this information to write the 2 × 2 transition matrix.
(b) Consider the case where we know today is nice, then the initial state matrix
will be: [1 0]. Find the probability the weather will be nice in three days.
(c) Assume the weather today is unpleasant. Find the probability the weather
will be nice in three days.

10. The annual Oxford - Cambridge boat HOMEWORK
race, has been rowed regularly since
1839. Using the data from 1839 up to To
1982, there were 58 Oxford wins and O C
67 Cambridge wins. If the relationship
between the results of a given year and From
the results of the previous year are
considered, the following table can be
constructed:
(a) Convert the “Number of wins” to percentages to rewrite the above matrix.
(b) If Oxford wins this year, what is the probability they will win next year?
in two years? three?
(c) Over many years, what percentage of games will Oxford win? Cambridge?
(d) Redo question (b) and (c) above assuming that Cambridge wins this
year. How do your answers to each question change?