Observing Representation
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Mr. Karsky: This is good work. Now let's take this problem up one more level. What is the probability of starting downtown and being downtown after a third trip? Explain your method.

Dewane: So all we really need to do is multiply the matrix times itself three times for three passengers or trips.

Mr. Karsky: How will we know what the answer is?

Jenny: Just use a calculator and multiply the matrix by itself three times, or to the third power, since each cab takes three fares. The answer about starting downtown and ending there again after three trips is in the middle, where the pickup downtown row crosses with the end downtown row. I tried it and got the same answer as the tree diagram, 0.309.

Mr. Karsky: How can you represent your method?

Louise: We can use the same original matrix, T, and find the solution matrix, S.

S = T•T•T = T3

Mr. Karsky: What about these other numbers in matrix S?
Anita: I don't think we need them, unless you make up another problem, because they tell about different starting or ending places.

Mr. Karsky: Nice work, everyone. Now let's discuss some connections between our matrix approach and the tree-diagram representation. Dewane, what do you think?

Dewane: Well, the initial values in the matrix were straight from the table of probabilities. The same numbers get multiplied and then added in both methods. Also, I noticed that the sum of any row was 1, just like the tree diagram. Probabilities have to add up to 1. It makes sense because in this problem, all the cabs starting from a location have to end up either in N, D, or S.

Anita: But the columns don't all add up to 1. I think that's because a column tells the probabilities of cabs ending up in places. They come from different groups of cabs, not from one whole group.

Susan: The same numbers are used in both ways of looking at it -- the tree diagram and the matrix -- and multiplied just the same.

Eric: But you don't see all the same in-between products in the matrix representation. They get added up automatically, and just the sum is written in the solution matrix.

Mr. Karsky: Wow, good observation. Now, to sum up what you've learned, for homework I'd like you to write about the benefits and drawbacks of each representation -- that is, the tree-diagram approach and the matrices approach -- and also what you think will happen if the number of trips we look at is 10, 20 or 100.

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