I think you’re getting a but mixed up here. Actually ‘hypothesis testing (HT)’ is used to test the ‘null hypothesis’. So when you said “HT is the use of statistics to determine the probability that a given hypothesis is true”, null hypothesis is the ‘given hypothesis’ which you are testing.
Let me give you an example. For instance, you’re trying to check whether ‘Men are Taller than women in a particular organisation’ (P.S: don’t judge me from the example :D)
So we will collect the data for say 100 students in the organisation and lets define a null hypothesis, say ‘Men are taller than women’.
The hypothesis testing will try to find evidence to disprove the null hypothesis. Generally every HT technique would give a ‘pvalue’ and thumbrule is that a ‘pvalue’ less than 0.05 means that null hypothesis is invalid. ‘pvalue’ is actually the probability of the observed data given the null hypothesis is true. If this probability is less than 0.05 (or some other threshold), then we reject the null hypothesis.
The last part about ‘pvalue’ is a bit tricky. Feel free to discuss further.