I am trying to understand how do legal systems handle situation where there is no clear causal responsibility of a person, but a well established stochastic relation. I am aware of the "beyond reasonable doubt" standard in some legal systems but I'd welcome any more information of text to continue research. For an example, let us assume that it's proven that certain chemical compound can induce a specific disease - some type of cancer, for an example. It is also well established that exposure to that chemical is not sole cause of this type of cancer, so there is incidence in a healthy population. So, let say that incidence doubles and it is found that a factory upstream of affected population has been dumping this chemical into water source. Person that has this disease sues the factory. There is no scientific way to prove that the person wouldn't get it without exposure, but it can be proven that this is probable (50%) and that the person was exposed. How would such situation be viewed in continental or common law?
As an academic matter, the US system has struggled with this, and law professors spend their lives debating it.
As a realistic matter, I'd bet that company in your example gets nailed.
On the theory, the issue here is but-for cause. Civil liability requires that the jury find it more likely than not, or over 50% likely, that but-for D's actions, P's injuries would not have occurred. Your hypo puts it at exactly 50, and tie goes to the defendant, so in this case even the theory is clear. But, say you change it a little bit to make it more like 65 or 70%. Now, you're getting into the territory of the classic blue bus case. A man is hit by a blue bus. Two companies in town own blue busses, and one company owns 90% of them. Can he recover against that company with no other evidence? The black letter answer is no, that naked statistical evidence doesn't cut it. The rationale is that if it were the other way, anytime someone got hit by a blue bus, that company would automatically have to pay out unless they had some evidence to disprove their liability. The burden of proof has been shifted, violating a fundamental tenet of our legal system.
But, this doctrine obviously poses an issue for plaintiffs like yours. It's one thing to say that someone hit by a bus needs to have more evidence. Theoretically, they could go out and get it: they could find the bus routes and schedules, examine where they were, and show that X company had a bus going by there at that time. They can also canvas for witnesses. It's much harder when, as your excellent hypo touches on, the science is so complicated. One controversial solution that has been debated a lot recently is market-share liability, but that doesn't seem pertinent here because you have mentioned another company.
On the legal realist side. As far as I can tell the system usually finds a way to hold the company responsible. One example is the tobacco litigation. My understanding is the science couldn't definitively prove that smoking caused cancer, but it could prove that it raised the risk significantly. Here's some information on that https://www.tortmuseum.org/the-tobacco-cases/ from the American Museum of Tort Law, which, yes, is a real thing.
As I understand your question, it is about evidence / proof and not legal liability. We can divide the issue into two parts, criminal law and tort law, since the standards of proof are different. In either case, there will be some question such as whether A caused the death or injury of B. In the US and common law jurisdictions in general, if this is a criminal offense, the state must prove beyond reasonable doubt that A caused the death or injury of B (that is a generic description of the standard, but jurisdictions differ in how they express the idea). In a tort action (where one party sues for damages), the standard is generally "more likely than not".
The disclaimer "there is no clear causal responsibility" tends to (a) limit this to tort cases and (b) support a finding in favor of the defendant, since they didn't clearly cause it. I assume you intend (a) and didn't intent (b). The intuitive difference between reasonable doubt vs. balance of probabilities is that with "beyond reasonable doubt", the evidence is so strong that there is no reason to reject the claim. "Reason" refers to "fact", not simple "you could imagine otherwise". With balance of probabilities, an individual developing cancer could (with reference to scientific facts) be from a dozen different reasons, but if it is more likely than not that the cancer was the result of A dumping benzine in the water, and that alternative reasons are less probable, then by that standard of proof, it is established that "A causes harm to B". The hard-science part comes in computing those probabilities, hence the need for expert witnesses who can speak to alternative explanations.