Transition Matrices

jump.in.motion

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.

An Example

The "Initial State Matrix" The "Transition Diagram"

The "Transition Matrix"

T = EUE

U

T =U

U

EE

S =E U

S =E U E U

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.

Predicting the future ...

After 1 Year

After 2 years

After 5 Years

After 50 years

Demonstrate stabilization ...

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.

(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%

small: 42%large: 58%

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?

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?

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.

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

(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.

(a) Use this information to write the 2 × 2 transition matrix.

HOMEWORK

The annual Oxford - Cambridge boat race, has been rowed regularly since 1839. Using the data from 1839 up to 1982, there were 58 Oxford wins and 67 Cambridge wins. If the relationship 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.

(d) Redo question (b) and (c) above assuming that Cambridge wins this year. How do your answers to each question change?

(c) Over many years, what percentage of games will Oxford win? Cambridge?

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

From

ToO C

HOMEWORK