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"If the hypothesis is that the coin comes up heads 95% of the time, then one time in twenty you will see what looks like contrary evidence."

My question here assumes that you mean one in twenty times you get a tails (if you mean one in twenty times you get a heads, then I'm also confused but for different reasons).

Surely if I have a hypothesis that a coin will land heads 95% of the time (and therefore tails 5% of the time) then every cluster of results in which 1/20 are tails is actually supporting evidence. If I toss a coin X times (where X is some number whereby 95% is a meaningful description of outcomes: X >= 20) and 1 out of those 20 is tails, that actually is solid evidence is support of my hypothesis - if, as you say "one in twenty times" I see a tails, that is very strong evidence that my 95% hypothesis is accurate...

Have I misread you point or am I thinking about this from the wrong angle?