Flips to collect two heads from a stingy coin

Question

A coin lands heads with probability 0.25 on each flip, independently. You flip it until you have seen 2 heads in total (they need not be consecutive). On average, how many flips does that take?

Accepted Answer

Collecting 2 heads is two independent waits for a single head stacked together: wait for the first head, then start fresh and wait for the second. The expected wait for one head is 1/p, so for r heads it is r/p by linearity of expectation: E = \frac{r}{p} = \frac{2}{0.25} = 8 	ext{ flips.} Takeaway. Expected flips to the r-th success is r/p; here 2/0.25 = 8.

Flips to collect two heads from a stingy coin

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