At the risk of over-explaining this, let’s ignore - just for a moment - the rarity of biased coins and consider possible results of 100 five-toss sequences for a biased, and an unbiased, coin:
This is not to say that the result is of no use. It does tend to prove the coin is biased. The strength of its tendency to prove bias is the likelihood ratio: the ratio of the probability of five-heads, given the coin is biased (from the above table this is 0.25) to the probability of five-heads, given the coin is unbiased (0.03), a ratio of 8.3 to 1. On the issue of bias, the result should be reported as: whatever the other evidence of bias may be, this result is 8.3 times more likely if the coin is biased than if it is not biased. The other evidence may be from a survey of coins which measured how often we can expect to find biased coins.
Again, this is not as surprising as it may seem at first glance. There may be only one biased coin in 10,000 coins, and one occurrence of five-heads from a biased coin in 40,000 coins (using the one-in-four frequency in the table), but, in round figures, there will also be 1200 occurrences (three per cent) of five-heads from unbiased coins in those 40,000 coins. This is why, on this occurrence of biased coins, a five-head result is much more likely (1200 times more likely) to be from an unbiased coin than from a biased one.
And, as an afterthought: if you feel estimating prior probabilities is a bit haphazard, the Bayesian formula can be turned around to tell you what priors you would need in order to get in the above example P(the coin is biased) = 0.95. You would, before doing the experiment, need to be convinced to a probability of about 0.70 that the coin was biased. This sort of approach is discussed in a paper by David Colquhoun (available courtesy of The Royal Society Publishing). If, as a lawyer, you want an easy introduction to Bayesian reasoning, see my draft paper on propensity evidence.