Retries under a hard cap

EasyExpected Value Stochastic Processes~4m

A trade-submission bot retries a flaky endpoint: each attempt independently succeeds with probability $0.4$ , and it stops the moment one succeeds. But it is allowed at most $3$ attempts - after a third failure it gives up. What is the expected number of attempts it makes? (to 2 decimals)

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

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1.96

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Retries under a hard cap

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