I don't understand what's wrong with this chain of logic.
- Null Hypothesis: A coin is fair
- Alternative Hypothesis: A coin is biased.
Suppose you flip a coin 900 times, and 450 times it comes up as heads. The probability of this is ~2%, so under a 5% significance level, we reject the null hypothesis.
More generally, I don't understand how hypothesis testing deals with events that are rare in general (e.g. getting exactly 450 heads). The null hypothesis seems to get rejected whenever the event is rare, even though the event can have maximum probability under the null hypothesis. In this case, the expected proportion of heads would be 0.5, which matches up with the null hypothesis.