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Students are explaining their results for two trips and drawing diagrams on the board.
Reflect on the problem and think about some questions you would ask this student. Then look at the questions listed below. For each question, think about an answer the student might provide; then select "Show Answer" to reveal a sample response.
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Mr. Karsky: How did you approach the problem, Reynaldo?
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Reynaldo: I made a tree, with the probabilities of being in one of the three destinations after starting downtown in the first column. Next, I put in the probabilities of leaving that destination and dropping off another passenger. I followed the paths that ended downtown and multiplied the two probabilities. Then I added the probabilities of all three ways of ending up downtown after two passengers.
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Mr. Karsky: That's an interesting approach. How did you handle multiple trips?
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Reynaldo: Well, I needed to extend the tree and calculate for ending downtown after three trips. So I drew another tree, and I realized I only needed to calculate for the third branches that ended downtown. I multiplied by the probability of going from any of the three areas to Downtown, like Northside to Downtown is 20%, and so on. Michael: The diagram seems confusing; it's like a maze.
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Mr. Karsky: Let's look at the meaning of just one of the paths through the maze, from the start, along the bottommost path in Reynaldo's diagram. Kim, what does that path represent?
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Kim: It says that if a cab starts downtown, there's a 50% chance that it will take a rider to Southside. Then, when it starts again in Southside, there's a 4/10 chance that it will end up somewhere in Southside again. When it takes the next rider, there's a 30% chance that passenger needs to go to Downtown.
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Mr. Karsky: Okay, Reynaldo has been mentioning probabilities. What is the maximum probability of any situation?
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Michael: One.
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Mr. Karsky: That's right, and is that represented in this diagram?
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Reynaldo: Yes. By looking at any level of the tree, the total of the possibilities adds up to exactly 1. Here, I can show you. 
Mr. Karsky: That's right, and a good thing in this case. Otherwise, it would mean some taxis got stuck somewhere other than in Northside, Downtown, or Southside!
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 Reflect on the student work
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