Probability that Null Hypothesis is True So, this may be a common question, but I’ve never found a satisfactory answer.  
How do you determine the probability that the null hypothesis is true (or false)?
Let’s say you give students two different versions of a test and want to see if the versions were equivalent.  You perform a t-Test and it gives a p-value of .02.  What a nice p-value!  That must mean it’s unlikely that the tests are equivalent, right?  No.  Unfortunately, it appears that P(results|null) doesn’t tell you P(null|results).  The normal thing to do is to reject the null hypothesis when we encounter a low p-value, but how do we know that we are not rejecting a null hypothesis that is very likely true?  To give a silly example, I can design a test for ebola with a false positive rate of .02: put 50 balls in a bucket and write “ebola” on one.  If I test someone with this and they pick the “ebola” ball, the p-value (P(picking the ball|they don’t have ebola)) is .02, but I definitely shouldn’t reject the null hypothesis that they are ebola-free.  
Things I’ve considered so far:


*

*Assuming P(null|results)~=P(results|null) – clearly false for some important applications.

*Accept or reject hypothesis without knowing P(null|results) – Why are we accepting or rejecting them then?  Isn’t the whole point that we reject what we think is LIKELY false and accept what is LIKELY true?

*Use Bayes’ Theorem – But how do you get your priors?  Don’t you end up back in the same place trying to determine them experimentally?  And picking them a priori seems very arbitrary.  

*I found a very similar question here: stats.stackexchange.com/questions/231580/. The one answer here seems to basically say that it doesn't make sense to ask about the probability of a null hypothesis being true since that's a Bayesian question.  Maybe I'm a Bayesian at heart, but I can't imagine not asking that question.  In fact, it seems that the most common misunderstanding of p-values is that they are the probability of a true null hypothesis.  If you really can't ask this question as a frequentist, then my main question is #3: how do you get your priors without getting stuck in a loop?


Edit:
Thank you for all the thoughtful replies.  I want to address a couple common themes.


*

*Definition of probability: I'm sure there is a lot of literature on this, but my naive conception is something like "the belief that a perfectly rational being would have given the information" or "the betting odds that would maximize profit if the situation was repeated and unknowns were allowed to vary".  

*Can we ever know P(H0|results)?  Certainly, this seems to be a tough question.  I believe though, that every probability is theoretically knowable, since probability is always conditional on the given information.  Every event will either happen or not happen, so probability doesn't exist with full information.  It only exists when there is insufficient information, so it should be knowable.  For example, if I am told that someone has a coin and asked the probability of heads, I would say 50%.  It may happen that the coin is weighted 70% to heads, but I wasn't given that information, so the probability WAS 50% for the info I had, just as even though it happens to land on tails, the probability WAS 70% heads when I learned that.  Since probability is always conditional on a set of (insufficient) data, one can never not have enough data to determine it and so it should always be (theoretically) knowable.
Edit: "Always" may be a little too strong.  There may be some philosophical questions for which we can't determine probability.  Still, in real-world situations, while we can "almost never" have absolute certainty, there should "almost always" be a best estimate.

 A: In order to answer this question, you need to define probability. This is because the null hypothesis is either true (except that it almost never is when you consider point null hypotheses) or false. One definition is that my probability describes my personal belief about how likely it is that my data arose from that hypothesis compared to how likely it is that my data arose from the other hypotheses I'm considering.  If you start from this framework, then your prior is merely your belief based on all your previous information but excluding the data at hand. 
A: The key idea is that, loosely speaking, you can empirically show something is false (just provide a counterexample), but you cannot show that something is definitely true (you would need to test "everything" to show there are no counterexamples).
Falsifiability is the basis of the scientific method: you assume a theory is correct, and you compare its predictions to what you observe in the real world (e.g. Netwon's gravitational theory was believed to be "true", until it was found out that it did not work too well in extreme circumstances).
This is also what happens in hypothesis testing: when P(results|null) is low, the data is contradicting the theory (or you were unlucky), so it makes sense to reject the null hypothesis. In fact, suppose null is true, then P(null)=P(null|results)=1, so the only way that P(results|null) is low is that P(results) is low (tough luck).
On the other hand, when P(results|null) is high, who knows. Maybe null is false, but P(result) is high, in which case you cannot really do anything, besides designing a better experiment.
Let me reiterate: you can only show that the null hypothesis is (likely) false. So I would say the answer is half of your second point: you don't need to know P(null|results) when P(results|null) is low in order to reject null, but you cannot say null is true it P(results|null) is high.
This is also why reproducibility is very important: it would be suspicious to be unlucky five times out of five.
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(edit: I think it would be useful to put a version of my comment to this question on top in this answer, as it is much shorter)
The non symmetric computation of p(a|b) occurs when it is seen as a causal relationship, like p(result|hypothesis). This computation does not work in both directions: a hypothesis causes a distribution of possible results, but a result does not cause a distribution of hypotheses. 
P(result|hypothesis) is theoretic value based on the causation relationship hypothesis -> result.
If p(a|b) expresses a correlation, or observed frequency (not necessarily a causal relation), then it becomes symmetric. For instance if we write down the number of games a sports teams wins/loses and number of games sports team scores less than or equal as / more than 2 goals in a contingency table. Then P(win|score>2) and P(score>2|win) are similar experimental/observational (not theoretic) objects.
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Very simplistic
The expression P(result|hypothesis) seems so simple that it makes one easily think that you can simply reverse the terms. However, 'result' is a stochastic variable, with a probability distribution (given the hypothesis). And 'hypothesis' is not (typically) a stochastic variable. If we make 'hypothesis' a stochastic variable then it implies a probability distribution of different possible hypothesises, in the same way as we have a probability distribution of different results. (but results does not give us this probability distribution of hypothesis, and merely changes the distribution, by means of Bayes theorem)

An example
Say you have a vase with red/blue marbles in a 50/50 ratio from which you draw 10 marbles. Then you can easily express something like P(outcome|vase experiment), but it makes little sense to to express P(vase experiment|outcome). Outcome is (on it's own) not the probability distribution of different possible vase experiments.
If you have multiple possible types of vase experiments, in that case it is possible to use express something like P(type of vase experiment) and use Bayes rule to get a P(type of vase experiment|outcome), because now the type of vase experiment is a stochastic variable. (note: more precisely it is P(type of vase experiment|outcome & distribution of type of vase experiments))
Still, this P(type of vase experiment|outcome) requires a (meta-)hypothesis about a given initial distribution P(type of vase experiment).

Intuition
maybe the expression below helps to understand the one direction
X) We can express the probability of X given a hypothesis about X.
thus
1) We can express the probability for results given a hypothesis about the results.
and
2) We can express the probability of a hypothesis given a (meta-)hypothesis about these hypothesises. 
It is Bayes rule that allows us to express an inverse of (1) but we need (2) for this, hypothesis needs to be a stochastic variable.

Rejection as solution
So we can not obtain a absolute probability for a hypothesis given the results. That is a fact of life, trying to fight this fact seems to be the origin of not finding a satisfactory answer. The solution to find a satisfactory answer is: accepting that you can not get a (absolute) probability for a hypothesis. 

Frequentists
In the same way as not being able to accept a hypothesis, we should neither (automatically) reject the hypothesis when P(result|hypothesis) is close to zero. It only means that there is evidence that supports change of our believes and it depends also on P(result) and P(hypothesis) how we should express our new believes.
When frequentists have some rejection scheme then that is fine. What they express is not wether a hypothesis is true or false, or the probability for such cases. They are not able to do that (without priors). What they express instead is something about the failure rate (confidence) of their method (given certain assumptions are true).

Omniscient
One way to get out all of this is to elliminate the concept of probability. If you observe the entire population of 100 marbles in the vase then you can express certain statements about a hypothesis. So, if you become omniscient and the concept of probability is irrelevant, then you can state wether a hypothesis is true or not (although probability is also out of the equation)
A: You have certainly identified an important problem and Bayesianism is one attempt at solving it.  You can choose an uninformative prior if you wish.  I will let others fill in more about the Bayes approach.
However, in the vast majority of circumstances, you know the null is false in the population, you just don't know how big the effect is.  For example, if you  make up a totally ludicrous hypothesis - e.g. that a person's weight is related to whether their SSN is odd or even - and you somehow manage to get accurate information from the entire population, the two means will not be exactly equal. They will (probably) differ by some insignificant amount, but they won't match exactly. 
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If you go this route, you will deemphasize p values and significance tests and spend more time looking at the estimate of effect size and its accuracy. So, if you have a very large sample, you might find that people with odd SSN weigh 0.001 pounds more than people with even SSN, and that the standard error for this estimate is 0.000001 pounds, so p < 0.05 but no one should care. 
