Two walks returning together

Two tokens perform independent simple symmetric random walks on the integers, starting at 00 and 44 respectively. After each takes 1010 steps, what is the probability that both are back at their starting positions? Round to four decimal places.

Show hints (2)+
  1. One walk returns after 1010 steps iff 55 right and 55 left: probability (105)/210\binom{10}{5}/2^{10}.
  2. The two walks are independent - square it.

Answer

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0.0606 (± 0.001)

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Asked at: Jane Street, Two Sigma

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