Finite Math B – Mrs. Leahy

10.1 Basic Properties of Markov Chains

Suppose two local television news networks go through a “ratings week” at the end of each month. They try to offer special segments that will draw viewers from the other station. During each period, Network A wins over 20% of Network B’s viewers, but loses 10% of its viewers to Network B.

Draw a graph to represent the movement between stations. This is called a TRANSITION DIAGRAM.

Draw a circle for each network.

Use arrows to represent the percentage

of viewers who change networks.

Use arrows to represent the percentage

of viewers who STAY the same.

Set up a TRANSITION MATRIX to represent the diagram.

Label the rows and columns “A” and “B”, one for each.

The position (row, column) represents the Probability of moving from the “row state” to the “column state.” Or, in this problem, the probability of moving from the row network to the column network.

PROPERTIES OF TRANSITION MATRIX:

1. It is a square matrix since all possible states must be used as rows and as columns.

2. All entries are between 0 and 1, inclusive, since all entries represent probabilities.

3. Each row must have a sum of 1, since the numbers represent the probabilities of changing from the state on the left to one of the states across the top.

Example 1: Which of the following could be a transition matrix (by definition)? Sketch a transition diagram for any transition matrix.

a) b) c)

(example 1 continued)

d) e)

Example 2: Write the transition diagram as a transition matrix.

a)

b)

Example 3: The British scientist Sir Francis Galton studied inheritance by looking at distributions of the heights of parents and children. In 1886 he published data from a large sample of parents and their adult children showing the relation between their heights. The following matrix is based on his data.

a) What percentage of “tall” parents had “tall” children?

b) What percentage of “short” parents had “medium” height children?

Use matrix multiplication, or a graphing calculator to find . This would represent another generation of children.

c) What proportion of “tall” parents had “tall”

grandchildren?

d) What proportion of “short” parents had “short” grandchildren?

Use matrix multiplication or a graphing calculator to find . This would represent 10 generations of children later.

e) What do you think these numbers represent?

In the “long run,” what proportion of people do we expect to be:

Tall? ______Medium Height? ______Short? ______

Example 4: Find the next two powers of each transition matrix (). Then find the indicated probability.

a) Find the probability that state 1 changes to state 2 after 3 repetitions.

(Example 4 continued)

b)

Find the probability that state 3 changes to state 1 after 1 time period.

Find the probability that state 2 changes to state 1 after 3 time periods.

Example 5: Find

c)

Find the probability that state 1 changes to state 5 after 1 repetition experiment.

Find the probability that state 1 changes to state 5 after 5 repetitions of the experiment.

Find the probability that state 3 changes to state 4 after 5 repetitions of the experiment.

Homework: pg 564-568 #10-18all, 20-26even, 28

10.1B Basic Properties of Markov Chains - Continued

Suppose two local television news networks go through a “ratings week” at the end of each month. They try to offer special segments that will draw viewers from the other station. During each period, Network A wins over 20% of Network B’s viewers, but loses 10% of its viewers to Network B.

Suppose Network A currently holds 60% of the “market share” of viewers. At the end of six months, what percentage of the market share will it hold?

A PROBABILITY VECTOR is a matrix of only one row that usually represents the initial state of the population… sometimes we call this the “market share.”

Probability Vector for our television networks:

We can PREDICT the probability vector/market share after a period of time using matrix multiplication.

Example 1:

Initial Probability Vector for news example:

Transition Matrix for news network:

“At the end of six months, what percentage of the market share will it hold?”

Example 2: If the current population has 20% tall, 40% medium, and 40% short people, what percentage of the population will be tall in 3 generations?

Example 3: A study is done to classify people by income as “Lower-class,” “Middle-class,” and “Upper-class.” For convenience we will call these “state 1, 2, and 3,” respectively. It is determined that, for example, the probability of children of state 1 (lower-class income) to also be state 1 as 0.65. The other probabilities are given in the following transition matrix.

After a census, it is determine that currently 21% of the population is state 1, 68% of the population is state 2, and 11% of the population is state 3.

What percentage of the population will be in state 1 in 3 generations?

Homework: p564-568 #1-9, 33abc, 37ab, 38abcd, 40abcd

ARENT??wing matrix is based on his data.
ple of parents and their adult children showing the relation between their heights.