Several times I was asked "wat betekent het eigenlijk, als we zeggen dat iets toeval is". I think there are many many answers, depending on the context. What is chance? Courts, scientists, ordinary people in everyday life, are continually seeing things which seem to have meaning, which seem to demand an explanation. But perhaps it is just chance. For instance, was it just chance that Joan Ginther won the jackpot in the lottery four times over the space of about 15 years? Could be, of course... But suppose you happened to learn that she had a PhD in statistics from Stanford university, lived in Las Vegas, was well off, did not seem to have any job, but just occasionally travelled to a small town with a huge amount of money strapped to her waist and bought all the tickets from the town store?
We tend to say that something is not due to chance when we believe there is a strong reason for it being the way it is and not different. Surely all those deaths and incidents always being in Lucia's shifts can't be due to chance? No: Lucia caused them, you could say. The hospital defined an incident to be a serious incident only if it occured in Lucia's shifts. And a death in the shift after Lucia's was considered to be the result of events in her shifts. Sure, Lucia was a direct cause of the extraordinary size of the correlation, but the reason for that, I believe, was that the hospital, for various other reasons, was already committed to believing she was a serial killer. One of the reasons was perhaps that she did have a bit of bad luck... not 1 in 342 million, but more like 1 in 100, I would guess.
Scientific researchers often want to prove that certain things are for real (eg, ESP, or the new drug makes people better, or left-handed people hae different life-span to right-handers). The usual appoach is to collect data on the question and look at it. Does it exhibit the pattern you would expect in the case that ESP exists, or whatever? Maybe the pattern seems to point in the direction you expect but could perhaps also just be due to chance. The scientist computes the probability that the size of the effect, or of the difference, of the sharpness of the pattern, would be as large as it actually happened to be, were in fact only chance at work. If that probability is smaller than 0.05 he or she will publish the results in a scientific journal saying that it is significant (at the 5% level), the university will put out a press-release, and the journalists will copy it.
Part of the problem here is to define what you do mean precisely by chance. For instance, in the Lucia case, what is the usual kind of variation in numbers of incidents in different nurses shifts, when there is no one around murdering babies? The fact is, that we hardly know! It has hardly ever been investigated. One thing for sure, is that it is not like the randomness of a perfect lottery. There are clusters of events, and shifts also follow regular patterns. With clusters and gaps in time. This makes the chance of extreme numbers of events in one person's shifts much larger than you would expect if incidents and deaths rained down from above purely at random. Taking no notice of what has already happened. In fact, there are so-called hidden confounders which affect both incidents and shifts. The season, the day of the week. One patient experiences many crises. They're not independent of one another always with exactly the same (small) probability every day.
If you are interested in the philosophical side of the meaning of chance I highly recommend "De zin van het toeval" by Ilya Maso (published by "Ambo"). Though some will find it a bit "zweverig".
But why should we need to explain or define chance in terms of other things? We always used to be happy imagining that time, mass, distance all truly and really exist out there, and don't need to be "explained" in terms of other things, aren't we? (Though cosmologists are now busy trying to explain how these things emerged from a physics without mass and space and time as we know it. Why should we have a need to "explain" chance? Couldn't it also be a fundamental ingredient of the universe? Personally, I tend to believe that, and moreover, I understand from psycho-linguistics that our brains are hard-wired by evolution to always believe there is an explanation for anything we see; we are also hard-wired with concepts of objects, agents, space and number. Chance was something which we became confronted with later and which we never quite could get our minds around. Humans are unhappy when uncertain. We don't know what to do and that costs us brain power, and our brains are already eating up 25% of our body's energy resources. We use rough heuristic rules for inference in the face of uncertainty which are manifestly incorrect and moreover make us believe we are much more certain of what we deduce (or induce) than is actually justified. Shoot first, ask questions afterwards. We are already statisticians by instinct (infering causes from effects), but we are bad statisticians. Similarly we are already psychologists by instinct (infering motives of others' actions). We do it well when in familiar surroundings but also our pychological inferences go wildly wrong in extreme situations. We look for patterns around us and act on them, we attribute motives to other people so we can predict or control them. We move in space and have instinctive understanding of relative directions.
Couldn't it have been different? Maybe elsewhere in the universe there are intelligent beings who evolved in an environment where chance was so significant that parts of probability theory got hard-wired into their brains. We don't have that.
This is all personal amateur philosophy, not science, not mathematics. A mathematical statistician or a probabilist knows the rules for how to do probability calculations (probabilities in, probabilities out). They don't need to know what probability means. Just as one can prove theorems of Euclidean geometry without an intuitive understanding of space and objects.
The applied statistician or probabilist does have an obligation to have an idea, what the meaning if of what he or she is doing. My own opinion is that when we use probability, we are saying that certain situations are analogous to situations which we understand well and moreover can idealize. A perfect lottery. A perfect toss of a perfect die. In an application one must say what is supposed to be fixed, what is supposed to be varying. Where does that variation come from. Are we taking account of all important sources of variation? Do all these sources behave at some level as if built up on top of a perfect lottery? When we use subjective probability too, we must be very careful to explain to the consumers whose uncertainty about fixed but unknown things out there we are putting into our calculations.
Probability theory consists of mathematical models for chance or uncertainty, and is used to compute the probabilities of effects given knowledge of the probabilities in the causes. Statistics goes a step further and tries to make deductions about causes from knowledge of (partlly random) effects. It therefore actually also contains a mathematical model for how we ought to make decisions, or how we ought to alter our beliefs, in the face of uncertainty. Again, that is an idealization! A simplification! Different models which emphasize different features of decision problems or of inference problems are possible. In particular there is a Bayesian paradigm and a frequentist paradigm. These are in my opinion two extreme models which each focus very stongly on one particular aspect and thereby do not do justice to another. Both are models. Models are tools we use to play with. We play with data by looking at it through the point of view of different models. There is not only one game in town. Playing several different games with the same opponent might teach you much more about him or her, than just one.