First tail within two flips

EasyProbability~4m

Each of $3$ people repeatedly flips the same biased coin, which shows heads with probability $p = 0.25$ , stopping at their first tails. What is the probability that none of the three needs more than two flips?

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One person needs $>2$ flips iff the first two are both heads: probability $p^2$ .
So one finishes in $\le 2$ with probability $1-p^2 = 15/16$ . Cube for three independent people.

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0.824 (± 0.001)

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Asked at: SIG, Akuna Capital

First tail within two flips

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