Earlier quoted context omitted.
Reverse isn't the right word here, either. Better to restate the definition: the p-value is the probability of observing the data (as represented by its aggregate statistics), under the condition of the null hypothesis.
I say 'reverse' because that's what it is formally: confusing P(data|null) with P(null|data). It reverses the conditional probability.
> If it is less than 5% probable that the null hypothesis is correct given our data, we will reject the null hypothesis and consider our alternate hypothesis (HA) to be correct.
You could be reversing the 5% to 95%, could be reversing null hypothesis to alternate hypothesis, could be reversing reject to accept, etc. Too many moving parts. P-values are so often misinterpreted, it's best to err on the side of specificity.