Crossing the origin twice

A simple symmetric random walk starts at 0 and takes steps of ±1\pm1. What is the probability that it is at 0 both after 2 steps and after 4 steps? Give a decimal to four places.

Show hints (2)+
  1. Steps 1–2 and steps 3–4 are independent blocks; each must net to 0.
  2. P(a 2-step block nets 0)=12P(\text{a 2-step block nets }0)=\tfrac12; multiply the two blocks.

Answer

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0.25

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Asked at: Probability & Market-Making, Timed Mental-Math & Sequences

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