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Possibility Trees Jeffrey Martinez Math 170 Dr. Lipika Deka 11/22/13

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Chapter 9.2 Problem #10 Suppose there are three routes from North Point to Boulder Creek, two routes from Boulder Creek to Beaver Dam, two routes from Beaver Dam to Star Lake, and four routes directly from Boulder Creek to Star Lake. (Draw a sketch.) a. How many routes from North Point to Star Lake pass through Beaver Dam? b. How many routes from North Point to Star Lake bypass Beaver Dam? North Point Boulder Creek Star Lake Beaver Dam

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Part (a) solution using a possibility tree: Part (a): We can see that there are several routes to Star Lake from North Point that go through Beaver Dam. Using our sketch, we see that in order to go through Beaver Dam we must go from Star Lake to Boulder Creek, then from Boulder Creek to Beaver Dam, and from Beaver Dam to North Point. We can make a possibility tree to find just how many unique routes there are. From Star Lake there are 3 different routes we can take to Boulder Creek, from Boulder Creek there are 2 different routes to Beaver Dam, from Beaver Dam there are 2 different routes to North Point. Using this info, we make our tree: From our tree we see that there are 12 unique routes that pass through Beaver Dam on the way to North Point. Star Lake Boulder Creek Beaver Dam North Point

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Solution using the Multiplication Rule How do we represent what we drew on the graph mathematically? We use the multiplication rule, which (from our textbook) states: If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n 2 ways [regardless of how the first step was performed], …the kth step can be performed in n k ways [regardless of how the preceding steps were performed], then the entire operation can be performed in n 1 n 2 · · · n k ways. (p. 527) Using our numbers from part (a), our process is a 3 step operation: Step 1 has 3 different ways it can be performed Step 2 has 2 different ways it can be performed Step 3 has 2 different ways it can be performed Using the multiplication rule, we find that there are 3∙2∙2 = 12 possible ways to go from North Point to Star Lake passing through Beaver Dam.

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Part (b) using the multiplication rule: Since we have determined in part (a) that this type of problem utilizes the multiplication rule, the solution for part (b) takes the same form as the solution for part (b). For clarification, let’s restate the problem: b. How many routes from North Point to Star Lake bypass Beaver Dam? Setting it up the same way as part (a), we see from our sketch that bypassing Beaver Dam makes part (b) a two step process. First we must travel from North Point to Boulder Creek, and from Boulder Creek we travel directly to Star Lake. From North Point to Boulder Creek there are still only 3 possible routes, or 3 possible ways it can be performed. From Boulder Creek to Star Lake there are 4 possible routes, or 4 possible ways it can be performed. Using the multiplication rule, we find that there are 3∙4 = 12 possible ways to travel to to Star Lake from North Point that bypass Beaver Dam.

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