Answer # 8
Question: The DERIVE command comb (n, r) simplifies to give you the number in position r of the nth row of Pascal's Triangle. Find the number in row 16 of Pascal's Triangle and use them to determine the average distance moved in a 16-step random walk. According to the relationship we obtained for the average length of a random walk of a given number of steps, how far would we expect to go, on the average, in a 16-step random walk.
Answer:
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The numbers in row 16 of Pascal's Triangle are (starting from left to right...not that it matters) - 16, 120, 560, 1820, 4386, 8008, 11440, 12870, 11440, 8008, 4386, 1820, 560, 120, 16, 1. |
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It is logical, that in a random walk, the chances of ending up closer to the middle are greater than the chances of ending up closer to the end. |
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In order to find how far you will go on average, you would would double the total number of steps (absolute value) and then multiply them by the number of steps away from the middle. Then, you would have to divide that number by 16^2. |
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The equation looks something like this: (2*16+32*14+240*12+1120*10+3640*8+8736*6+16016*4+22880*2+12820*0)/65536 |
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205920/65536=3.14208984375 |
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The average number of steps would probably be zero, or very near the middle, since the chances are a lot greater of landing there than landing anyone else. |
-Yash Patel