Spoiler for Suggestive proofThe results of a biased coin can be separated into fair-coin flips and two-headed coin flips.

Suppose

(H) = 0.6, so thatpp_{min}= 0.4.Representative flips are T T T T H H H H H H.

Partition them as: T T T T H H H H

||H H.2

p_{min}(80%) of the flips are "fair" flips (equal chance of T and H.) The others are "two-headed" flips.The fair flips reach H=T with

=1; the others reach H=T withp=0.p

Thus, with probability 2

p_{min}, H=T will eventually be reached.

Putting it differently: a biased coin acts with probability 2

p_{min}like a fair coin.

Woah.