Linking the Small PLINKO Board and Pascal’s Triangle: Student Discussion Guide Examine the number of paths from starting SLOT B. Draw in all the possible paths a chip can take from B to each ending slot. The paths to SLOT A’ are drawn for you. How many possible paths are there to A’? 3 B’? 6 C’? 3 How does this differ from our earlier probability calculations using the tree diagram/formula? The numbers do not match the numerators of the probability fractions. How well might this process or using a tree diagram work for the large PLINKO board? Why would it or wouldn’t work well? Not well. It would be difficult, if not impossible, to keep track of all of the different paths. How might we use a more efficient way to count the number of possible paths to get to each prize level? Look for patterns to find a mathematical way of counting/computing the total paths. Let’s look at the board a little closer. We can write in the number of paths to get to a certain region in each cell. Then, look for patterns. How were each of these numbers determined? By counting the total number of ways to get to each “diamond cell.”You can also add the two numbers above a cell to get the value for that cell. What patterns do you notice? Answers vary. For most, you add the two number above the cell to get the one below. The exception is the edges. The left and right side are the same (i.e., 1, 2, 1; 3, 6, 3) Are there really only 12 possibilities we need to consider when determining theoretical probability? Why or why not? Consider examining your tree diagram/formula for SLOT B as well.
Linking the Small PLINKO Board and Pascal’s Triangle: Student Discussion Guide Examine the number of paths from starting SLOT B. Draw in all the possible paths a chip can take from B to each ending slot. The paths to SLOT A’ are drawn for you. How many possible paths are there to A’? 3 B’? 6 C’? 3 How does this differ from our earlier probability calculations using the tree diagram/formula? The numbers do not match the numerators of the probability fractions. How well might this process or using a tree diagram work for the large PLINKO board? Why would it or wouldn’t work well? Not well. It would be difficult, if not impossible, to keep track of all of the different paths. How might we use a more efficient way to count the number of possible paths to get to each prize level? Look for patterns to find a mathematical way of counting/computing the total paths. Let’s look at the board a little closer. We can write in the number of paths to get to a certain region in each cell. Then, look for patterns. How were each of these numbers determined? By counting the total number of ways to get to each “diamond cell.”You can also add the two numbers above a cell to get the value for that cell. What patterns do you notice? Answers vary. For most, you add the two number above the cell to get the one below. The exception is the edges. The left and right side are the same (i.e., 1, 2, 1; 3, 6, 3) Are there really only 12 possibilities we need to consider when determining theoretical probability? Why or why not? Consider examining your tree diagram/formula for SLOT B as well.
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