Lets say there is a coin being flipped, and someone thinks that the coin is biased so that the probability of getting heads is less than 1/2.
If they flip a coin 10 times, and get heads twice, I understand why hypothesis testing is useful, but I don't understand why to test the hypothesis, we find the probability of getting less than or equal to two heads.
Why don't we just find the probability of getting 2 heads, and decide if that is significantly unlikely or not?
Intuitively, I kind of get that as the number of flips increases, the probability of any single number of heads becomes incredibly small, and so it would be absurd to focus on just that probability, but I still would like some help understanding why we consider the cumulative probabilities.
Thanks
A test sees whether the data is too "surprising" for us to still find the null hypothesis credible. The basic idea is we're working out the probability, conditional on the null hypothesis, something at least this surprising would happen. Richard Dawkins once coined the wonderful term petwhac, "population of events that would have appeared coincidental", to describe the set of events whose probability of containing what actually happened is relevant.
The probability something exactly this surprising would happen doesn't tell you anything. It's especially obvious that approach is unworkable for a continuous probability distribution, for which the exactly-this-surprising probability is $0$ for any data. But even in the discrete case, there is e.g. very little chance of exactly a million heads when tossing a fair coin two million times, but you wouldn't claim such an outcome is evidence against its being fair.