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Caspian comments on Too good to be true - Less Wrong
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Well, perhaps a bit too simple. Consider this. You set your confidence level at 95% and start throwing a coin. You observe 100 tails out of 100. You publish a report saying "the coin has tails on both sides at a 95% confidence level" because that's what you chose during design. Then 99 other researchers repeat your experiment with the same coin, arriving at the same 95%-confidence conclusion. But you would expect to see about 5 reports claiming otherwise! The paradox is resolved when somebody comes up with a trick using a mirror to observe both sides of the coin at once, finally concluding that the coin is two-tailed with a 100% confidence.
What was the mistake?
One mistake is treating 95% as the chance of the study indicating two-tailed coins, given that they were two-tailed coins. More likely it was meant as the chance of the study not indicating two-tailed coins, given that they were not two-tailed coins.
Try this:
You want to test if a coin is biased towards heads. You flip it 5 times, and consider 5 heads as a positive result, 4 heads or fewer as negative. You're aiming for 95% confidence but have to get 31/32 = 96.875%. Treating 4 heads as a possible result wouldn't work either, as that would get you less than 95% confidence.