(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.

#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.

#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).

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)

#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)