A study on the probability issue of“a frog jumping one meter a time”

Tianbo Qi The High School Attached to Tsinghua University Yintai Zhang The High School Attached to Tsinghua University

S.-T. Yau High School Science Awarded Papers mathscidoc:1608.35136

Dongrun-Yau Science Award, 2013
The problem we study in this paper originated from a simple high school math competition (AMC 12) question on a geometric probability model: “A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog’s final position is no more than 1 meter from its starting position?” [1] From here, we changed the number of jumps the frog made, the length of each jump and the dimensions of the space. Using recursion, we obtained the recursive formula of the probability of a frog landing x meters within its original spot after jumping m 1-meter jumps in an N -dimension space by integrating the probability distribution of the former step. We also suggested a concept “the intensity of the probability field” to describe the relative probability of each spot and gave an expression of the intensity of each spot of a 2-dimensional space when the frog makes m jumps.
probability, geometric probability model, function, N -dimensional space
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@inproceedings{tianbo2013a,
  title={A study on the probability issue of“a frog jumping one meter a time”},
  author={Tianbo Qi, and Yintai Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818134926173817228},
  booktitle={Dongrun-Yau Science Award},
  year={2013},
}
Tianbo Qi, and Yintai Zhang. A study on the probability issue of“a frog jumping one meter a time”. 2013. In Dongrun-Yau Science Award. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818134926173817228.
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