Monday, February 12, 2007

Transition Matrix

In the beginning of todays class, we had a review on last weeks lesson about multiplying matrices. Transition matrix was then introduced to us. Transition matrices involves probabilities and percentages. It describes change and can be symbolized with "[T]" To be able to work with these matrix, the columns and rows must have the same information.

This example I'm about to give is from the first transition matrices worksheet Mr K handed out. We had a hard time understanding example 2, which was the sports one, so ill re-cap on that one..

The annual Oxford-Cambridge boat race, has been rowed regularly since 1839. Using the data from 1839 up to1982, 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.


**-This means that Oxford got 35 games won after another (two games won in a row) and 22 wins without a two game in a row streak ... This goes the same for Cambridge.

To be able to work with this matrix, we need to convert to a percentage then to a decimal. I'm sure we all know how to convert these into decimals. the result should be.

Question: If Oxford wins this year, what is the probability they will win next year? in two years? in three three years?

Answer: to solve this question we need to have a transition matrix and a state matrix.

The transition matrix is :

This matrix represents the change. for example, there is a 61% chance that Oxford will have a two game winning streak.

The state matrix is simply stated "how it begins". For example, in the question it stated that Oxford won this year

Therefore we put a 1 under the column of Oxford in a 1x1 matrix which the rows represent the win. There is a zero under Cambridge because this represents the first year and they lost the first year.

We then multiply the transition matrix and the state matrix to get the probability of each team.
It goes:


The Transition matrix has an exponent of two which will give us the result for a second year percentage. We all know by now how to multiply matrices. That is the next step. the result is:

This means that in the next year. Both teams will have the same chance of winning between themselves.


.i hope this is much better than my other post.. :)

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