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Monday, February 13, 2006 

Transition Matrices

Hey, My apologizes for posting up so late!

But let's just get to the point. For today we had two periods of math, but in the morning class we were told that we were going to have an assembly. So that pretty much took up the whole morning class to learn anything. But Mr. Kuropatwa did give us a little chit chat on whats up with the blog. Like how he posted a puzzle/game yesterday and how it may also be math related. Mr. Kuropatwa will be doing this on every Sunday for now on. So be sure to check them out!

He also mentioned that he will be planning to set up a little chat box so that if we need help with anything we can ask questions and of course we can chat with each other?! Hopefully! But like what Mr. Kurapatwa said don't mention your name, your location, your school, or any kind of information that can give you away. So if you haven't read "Students Made This!", I advise you to do so. It talks about everything you should know when being online. It's pretty helpful I say.

As for our second class, we learned the last type of matrix in this unit. TRANSITION MATRICES.

Transition Matrix: a square matrix that shows how the probability of one event is dependent on the probability of another event occurring.

For example, in class we did this one example with sports. I know you guys probably have the notes on this already but for the ones who missed today's class but maybe if I explain it might help or not? I'll give it a shot.

The annual Oxford-Cambridge boat race, has been rowed regularly since 1839. Using the date 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:

From Oxford [35 22 ]
Cambridge [23 44 ]

So now the question askes us (a) to convert the "number of wins" to percentages to rewrite the above matrix.

So what we do is each score and divide it by the total of races.

[35/57 22/57]
[23/67 44/67]

We get 57 and 67 by adding each of the rows.

We then get this [.614 .3859]
[.3432 .6567]

(b) If Oxford wins this year, what is the probability they will win next year? In two years? Three?
So then to get to this solution we will take our STATE MATRIX which is [1 0] because it represents 1 for Oxford wins and 0 for Oxford loses. We take that and multiply it with our TRANSITION MATRIX which is our matrix up there where we figured our our percentages.

[1 0]x[.614 .3859]
[.3432 .6567] = [.614 .386]

In the resultant matrix in A11 it represents the percentage that Oxford wins. And in A12 that represents the percentage that Oxford loses.

And if you want to find out for the next year all you have to do is take the resultant matrix [.614 .386] and multiply it with our transition matrix.

[.614 .386]x[.614 .3859] =for the next year.
[.3432 .6567]

And if you want to find out for long term to see what percentages of games will Oxford win. All you have to do is take our state matrix with was [1 0] and multiply it with our transition matrix to the exponent of how ever much what to see it. Like say if you want to see if Oxford would still be winning after 10 years you'll just multiply it to the exponent of 10.

[1 0 ]x [.614 .3859] ^10=
[.3432 .6567]

Yeah, so I'm not to sure about this, but I tried...at least. So sorry guys!

Uhm, tomorrow's scribe will be....Dom.


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Sorry Im gonna have to put up the post later

Corrie, this is probably one of the most difficult topics of this unit. It's really hard to explain -- but you did a great job!

I really like the idea of using an example we discussed in class to explain it. I also like that you address it to classmates who may have missed class.

A couple of suggestions:

You wrote:
STATE MATRIX which is [1 0] because it represents 1 for Oxford wins and 0 for Oxford loses.

which is good but it should end:
1 for Oxford wins (because they did) and 0 for Cambridge wins (because they lost).

I also noticed everyone is having trouble getting all their matricies to line up. Try typing in your matricies in between this HTML code:

put your matrix here
spaced any way you like
it will post exactly
they way you type it here

Also, to write exponents, like x^2 we can use this HTML:


For subscripts as in the address of a matrix entry like A12, we use this HTML:


I hope that helps everyone who has been having these sort of problems.

Corrie, you did a really good job on this post. Well done!

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