If there is a belief for which there is an equal amount of evidence both for and against (the evidence is provided by experts in the appropriate field), what is the most rational thing to do in a situation like this?
In the case of equal evidence for and against a belief, I suggest the following considerations are relevant.
Is the equal evidence merely the evidence you have to hand ? The requirement of total evidence is unrealistic but if you have reason to believe that more evidence is available on further investigation then rationally you will look for that extra evidence. But this is hardly a clinching point, just a reminder of the obvious.
If you believe that you have all available evidence, then a resort to probability is appropriate. Call the belief 'X'. The available evidence for and against X, taking X as a discrete, singular belief, is equal. But you have other beliefs. No-one can have just one belief. Given your set of background beliefs - beliefs other than X - which is more probable, X or ¬X ? Given your background beliefs, Θ, is the probability of X being true greater than the probability of its being false ?
To see how this might work in practice : a ghost story is related to you. You have all the relevant details, know the people involved, can find no trace of fraud. One set of normally reliable people with no motive for deception assures you that a ghost was seen. Another group, same number of people and equally normally reliable and without motive for deception, assures you that there was no ghost-sighting. At one level you're stuck. But you have other beliefs, background beliefs. Suppose you are a physicalist whose view of the world excludes supernatural phenomena. Against your background beliefs it is more probable that no ghost was seen than that one was.
If you have different background beliefs, you may decide to believe it more probable that a ghost was seen. What it is rational to believe depends on the degree of justification - here, probability - one belief receives, or does not receive, from other beliefs. After all, a belief can only be rational in the light of other beliefs.
Equal evidence for and against a belief - which to choose?
The choice rests on (1) its potential consequences and (2) the decision-maker's tolerance of risk. In the question, the chooser is left with a coin toss, literally 50/50, about the true situation. An unguided fork in the road. So some other factor has to settle the question.
(1) What might happen? A loss of ten dollars? A losing candidate declared the winner? A defendant wrongfully going to jail? A bad grade on a biology exam?
(2) Can the chooser tolerate that outcome? Go through the same four questions. How willing is the chooser to risk the negative outcome on a coin toss?
And that is all you've got when you're facing a 50/50 probability of truth or falsehood.
People will have different beliefs. E.g. : Belief in the beings in another heavenly body, Belief in the influence of evil spirits, Belief in the influence of God.
Since the first belief usually does not affect us you can (we often) discard both beliefs (both for and against). If you are an explorer or a scientist who likes exploration, instead of believing that there are heavenly beings, you should explore and find out the truth.
Belief in the influence of evil spirit (the second one) can be discarded by analyzing more facts.
But the controversial one, the third one, of all eras can only be confirmed by realization by oneself. Otherwise we have to believe wise men's words. There is no other solution for this because when we check many other facts we will get equal number of counter arguments. That is why it is always a controversy. So, in this case the best way we can adopt is to seek throughout life by keeping ourselves clean always (in all aspects). If God can influence he should influence you also. There is no harm in it.
So, the third one is a crucial one. It is crucial because it can change our behavior and activities a great deal. Both beliefs (as a theist and as an atheist) can create overconfidence in both persons and eventually into a total perish. When this overconfidence grow in collective form it will be a great disaster. Most people forget this truth. We can see this in this present world. Since the failure in mutual understanding can grow into severe problems, Theists and Atheists must know one another.
Now in general case:
All the needles made from the same machine seems to have the same length. But there must be slight difference in their length. similarly, even though the two beliefs seemed to be equal amount of evidences (both for and against), there must be some difference; whatever you may argue. So one must be better.
For a better life, the humanitarian value must prevail in this world. Nobody can deny this. So I mean, in many occasions you must stick to the one that is useful to the humanity.
The apt word I should use is, 'Dharma'. But it may confuse people who believe in different philosophies and sometimes very difficult to decide which one is Dharma in one particular context. That was why I didn't use that word. Dharma can encompass/be the superset of 'actions based on humanity' (not always).
Since this question is about belief, we need not make much effort by thinking about Dharma. But when we have to act we should give the most preference to Dharma. I don't have the opinion that the actions based on humanity is always the subset of Dharma. I would say, it is a diluted form of Manava Dharma (implies Human Dharma).
Dharma is scientific. Why?
Let's take the following example:
Suppose, in order to look after his father who is in a very critical condition a popular singer who promotes humanitarian values cancels his music program. By canceling his program he is actually losing a chance for promoting humanitarian values. Here, if he thinks only of humanity he can't cancel his program. If he, the only son of his father, gave more importance to Putra Dharma (implies Son's Dharma) he will certainly cancel his program.
On the other hand, if he hadn't canceled that program he might not be able to sing well in that program and this may lessen his reputation. The feeling of guilt [since he didn't help his father at the time of his (father's) last breath] might haunt him until his death and this might affect his career. If he were with his father, he would certainly have confidence and it would boost up his ability in future. This is why I said I would always prefer Dharma.
Read about different types of Dharma. If this is purely unscientific you can ignore this. [I am sorry, I couldn't find a good explanation in other websites.] So, if scientific you can accept it.
tarko apratisthah srutayor vibhinna
nasau munir yasya matam na bhinnam
dharmasya tattvam nihitam guhayam
mahajano yena gatah sa panthah [Cc. Madhya 17.186]
Meaning: So dharma, the path of religiosity, is very confidential. Dharmasya tattvam nihitam guhayam. Then how I shall accept what is dharma, what is religion? Mahajano yena gatah sa panthah. You just follow the footsteps of authorized persons. Then you understand what is dharma. You cannot manufacture. So, here is the same system, the Vedic system is the same. Either you hear from the direct Vedas or scriptures following the Vedas.