More about transition matrices.

1. Transition Matrices
jump.in.motion

2. 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.

3. (c) Assume the weather today is unpleasant. Find the probability the
weather will be nice in three days.

4. (b) If Oxford wins this year, what is
the probability they will win next
year?
three?
in two years?
(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?

5. The annual Oxford - Cambridge boat
race, has been rowed regularly since
To
1839. Using the data from 1839 up to
O C
1982, there were 57 Oxford wins and
67 Cambridge wins. If the relationship
From
between the results of a given year
and 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?

6. Suppose that the population of a small island is classified into three
distinct groups, children, teenagers and adults, and that each year:
• children are born at a proportion of • 1 % of the children die
6% of the adult population
• 10 % of children become teenagers • 5 % of the teenagers die
• 14% of teenagers become adults • 7 % of the adults die
This year, in Youngville, a city with a population of 25 000, there
are 5 000 children, 18 000 teenagers and 2 000 adults.
(a) Write a row matrix that represents the current population of
Youngville.
(b) Write a 3 × 3 transition matrix that shows how the population
is changing over time.
(c) Find the population of each group in 10 years from now.

7. This year, in Youngville, a city with a population of 25 000, there
are 5 000 children, 18 000 teenagers and 2 000 adults.
(a) Write a row matrix that represents
the current population of Youngville.
• children are born at a proportion of • 1 % of the children die
6% of the adult population
• 10 % of children become teenagers • 5 % of the teenagers die
• 14% of teenagers become adults • 7 % of the adults die
(b) Write a 3 × 3
transition matrix
that shows how the
population is
changing over time.

8. (c) Find the population of each group in 10 years from now.

9. If a train is late on one day, there is a 90% probability that the same train
will be on time the next day, while if the train is on time, there is a 20%
chance it will be late the next day. If in a given week it arrives on time on
Monday, compute the probabilities that it will be on time or late for each
of the subsequent days of the week. What would the corresponding
probabilities have been if the train had been late on the Monday?
HOMEWORK

10. HOMEWORK
To reduce traffic congestion a city planner proposed the conversion of
some downtown streets to one-way streets. The new plan is below.
Water Park
Museum City Hall
Library
Tourist Centre
Court House
Art Gallery
(a) Construct a network matrix to represent the new plan.
(b)Determine the number of ways a tourist could get from the Art
Gallery to the Tourist Centre and see exactly 3 sites along the way.

11. HOMEWORK
Solve for x and y.