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99% confidence means you are accepting a 1% error on the ideal.
95% means that you are accepting a 5% error on the ideal.
It would be hard to get within 1% or 5% of the ideal with very few tosses. If you do a few tosses you may actually conclude that the coin is biased even if it's not. I suspect that within 10 or 20 tosses you should seriously approach your ideal 50-50.
With 4 coin tosses, HHHH has a probability of 6.25% which allows you to reject at 90% but not 95%.
With 5 coin tosses, HHHHH has a probability of 3.125% which allows you to reject at 95% but not at 99%.
The point is that, with less than 4 coin tosses you cannot show bias, no matter what. Sometimes you simply don't have enough data to argue statistically.
And statistics can only suggest where to look for actual evidence, it can't prove (or disprove) bias all by itself.
For instance, the contingency table analysis I did yesterday suggests looking for actual (not statistical) evidence of bias in the duration or the trials, not in the result. JakeS posted a theory that indictments were issued in the hopes of gathering sufficient evidence by the time the cases came to trial, which in some cases hasn't happened, resulting in prolongued imprisonments without trial rather than dismissals for lack of evidence. But a theory consistent with statistical suggestions is not evidence.
Most economists teach a theoretical framework that has been shown to be fundamentally useless. -- James K. Galbraith
The ideal result should be heads 50% of the time and tails 50% of the time.
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