Transitions ...... changing from one state to another.

Changing Weather

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

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.

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.

(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.

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.

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

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

The annual Oxford - Cambridge boat race, has been rowed regularly since 1839. Using the data from 1839 up to 1982, there were 57 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.

From

ToO C

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:

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

From

ToO C

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:

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

From

ToO C

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:

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

From

ToO C

HOMEWORK

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

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

(b) Write a 3 × 3 transition matrix that shows how the population is changing over time.

(a) Write a row matrix that represents the current population of Youngville.

This year, in Youngville, a city with a population of 25 000, there are 5 000 children, 18 000 teenagers and 2 000 adults.

HOMEWORK

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