Expected flips for three heads

A biased coin shows heads with probability 14\tfrac14 on each flip. What is the expected number of flips to observe 33 heads (not necessarily consecutive)?

Show hints (2)+
  1. The wait for one head is geometric with mean 1/p1/p.
  2. Three (non-consecutive) heads is three independent such waits: 3/p3/p.

Answer

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12

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Asked at: Optiver, SIG

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