Attempts to land three good data downloads

Question

A flaky server delivers each requested data file successfully with probability $0.5$ , independently each time. You need $3$ good downloads before you can start a job, and you keep requesting until you have them.

On average, how many requests does it take to collect the third successful download?

Accepted Answer

Waiting for the 3rd success is just waiting for the 1st success, done three times back to back. After each success the process resets and you again wait for one more success, independently. The expected wait for a single success with probability p is 1/p. To collect r successes you stack r of these independent waits, so by linearity of expectation E[	ext{attempts for } r 	ext{ successes}] = r \cdot \frac{1}{p} = \frac{r}{p}. Here r = 3 and p = 0.5: E = \frac{3}{0.5} = 6 	ext{ requests.} Takeaway. The expected number of trials to reach the r-th success is r/p: just r copies of the single-su…

Attempts to land three good data downloads

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