Retries under a hard cap

A trade-submission bot retries a flaky endpoint: each attempt independently succeeds with probability 0.40.4, and it stops the moment one succeeds. But it is allowed at most 33 attempts — after a third failure it gives up. What is the expected number of attempts it makes? (to 2 decimals)

Show hints (2)+
  1. A truncated geometric: the uncapped mean 1/p1/p is too large because the cap chops the tail.
  2. Use E[N]=k1P(Nk)=1+q+q2E[N]=\sum_{k\ge1}P(N\ge k) = 1 + q + q^2 with q=0.6q=0.6.

Answer

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1.96

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Asked at: Jane Street, Akuna Capital

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