[summary: What does it mean to say that a fair coin has a 50% probability of landing heads? There are three common views:

• The propensity interpretation: Some probabilities are just out there in the world. It's a brute fact about coins that they come up heads half the time; we'll call this the coin's physical "propensity towards heads." When we say the coin has a 50% probability of being heads, we're talking directly about this propensity.
• The frequentist interpretation: When we say the coin has a 50% probability of being heads after this flip, we mean that there's a class of events similar to this coin flip, and across that class, coins come up heads about half the time. That is, the frequency of the coin coming up heads is 50% inside the event class (which might be "all other times this particular coin has been tossed" or "all times that a similar coin has been tossed" etc).
• The subjective interpretation: Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it's already landed either heads or tails. The fact that I don't know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim "I think this coin is heads with probability 50%" is an expression of my own ignorance, which means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.

When the event in question is deterministic, the propensity view is commonly considered an example of the Mind projection fallacy. The frequentist view is self-consistent, but says that one-off events (such as the probability of it raining where you are tomorrow in particular) cannot be assigned "probabilities", because there are no objective frequencies to draw upon. The subjectivist view extends the frequentist view, and allows you to also consider probabilities (interpreted as betting odds) on one-off events. Probability theory provides tools for reasoning about subjective probabilities in a rigorous, consistent way.]

What does it mean to say that a flipped coin has a 50% probability of landing heads?

Historically, there are two popular types of answers to this question, the "[frequentist_probability frequentist]" and "subjective" (aka "Bayesian") answers, which give rise to radically different approaches to experimental statistics. There is also a third "[propensity propensity]" viewpoint which is largely discredited (assuming the coin is deterministic). Roughly, the three approaches answer the above question as follows:

• The propensity interpretation: Some probabilities are just out there in the world. It's a brute fact about coins that they come up heads half the time. When we flip a coin, it has a fundamental propensity of 0.5 for the coin to show heads. When we say the coin has a 50% probability of being heads, we're talking directly about this propensity.
• The frequentist interpretation: When we say the coin has a 50% probability of being heads after this flip, we mean that there's a class of events similar to this coin flip, and across that class, coins come up heads about half the time. That is, the frequency of the coin coming up heads is 50% inside the event class, which might be "all other times this particular coin has been tossed" or "all times that a similar coin has been tossed", and so on.
• The subjective interpretation: Uncertainty is in the mind, not the environment. If I flip a coin and slap it against my wrist, it's already landed either heads or tails. The fact that I don't know whether it landed heads or tails is a fact about me, not a fact about the coin. The claim "I think this coin is heads with probability 50%" is an expression of my own ignorance, and 50% probability means that I'd bet at 1 : 1 odds (or better) that the coin came up heads.

For a visualization of the differences between these three viewpoints, see Correspondence visualizations for different interpretations of "probability". For examples of the difference, see Probability interpretations: Examples. See also the Stanford Encyclopedia of Philosophy article on interpretations of probability.

The propensity view is perhaps the most intuitive view, as for many people, it just feels like the coin is intrinsically random. However, this view is difficult to reconcile with the idea that once we've flipped the coin, it has already landed heads or tails. If the event in question is decided [determinism deterministically], the propensity view can be seen as an instance of the Mind projection fallacy: When we mentally consider the coin flip, it feels 50% likely to be heads, so we find it very easy to imagine a world in which the coin is fundamentally 50%-heads-ish. But that feeling is actually a fact about us, not a fact about the coin; and the coin has no physical 0.5-heads-propensity hidden in there somewhere — it's just a coin.

The other two interpretations are both self-consistent, and give rise to pragmatically different statistical techniques, and there has been much debate as to which is preferable. The subjective interpretation is more generally applicable, as it allows one to assign probabilities (interpreted as betting odds) to one-off events.

Frequentism vs subjectivism

As an example of the difference between frequentism and subjectivism, consider the question: "What is the probability that Hillary Clinton will win the 2016 US presidential election?", as analyzed in the summer of 2016.

A stereotypical (straw) frequentist would say, "The 2016 presidential election only happens once. We can't observe a frequency with which Clinton wins presidential elections. So we can't do any statistics or assign any probabilities here."

A stereotypical subjectivist would say: "Well, prediction markets tend to be pretty [well_calibrated well-calibrated] about this sort of thing, in the sense that when prediction markets assign 20% probability to an event, it happens around 1 time in 5. And the prediction markets are currently betting on Hillary at about 3 : 1 odds. Thus, I'm comfortable saying she has about a 75% chance of winning. If someone offered me 20 : 1 odds against Clinton — they get \$1 if she loses, I get \$20 if she wins — then I'd take the bet. I suppose you could refuse to take that bet on the grounds that you Just Can't Talk About Probabilities of One-off Events, but then you'd be pointlessly passing up a really good bet."

A stereotypical (non-straw) frequentist would reply: "I'd take that bet too, of course. But my taking that bet is not based on rigorous epistemology, and we shouldn't allow that sort of thinking in experimental science and other important venues. You can do subjective reasoning about probabilities when making bets, but we should exclude subjective reasoning in our scientific journals, and that's what frequentist statistics is designed for. Your paper should not conclude "and therefore, having observed thus-and-such data about carbon dioxide levels, I'd personally bet at 9 : 1 odds that anthropogenic global warming is real," because you can't build scientific consensus on opinions."

…and then it starts getting complicated. The subjectivist responds "First of all, I agree you shouldn't put posterior odds into papers, and second of all, it's not like your method is truly objective — the choice of "similar events" is arbitrary, abusable, and has given rise to p-hacking and the replication crisis." The frequentists say "well your choice of prior is even more subjective, and I'd like to see you do better in an environment where peer pressure pushes people to abuse statistics and exaggerate their results," and then down the rabbit hole we go.

The subjectivist interpretation of probability is common among artificial intelligence researchers (who often design computer systems that manipulate subjective probability distributions), wall street traders (who need to be able to make bets even in relatively unique situations), and common intuition (where people feel like they can say there's a 30% chance of rain tomorrow without worrying about the fact that tomorrow only happens once). Nevertheless, the frequentist interpretation is commonly taught in introductory statistics classes, and is the gold standard for most scientific journals.

A common frequentist stance is that it is virtuous to have a large toolbox of statistical tools at your disposal. Subjectivist tools have their place in that toolbox, but they don't deserve any particular primacy (and they aren't generally accepted when it comes time to publish in a scientific journal).

An aggressive subjectivist stance is that frequentists have invented some interesting tools, and many of them are useful, but that refusing to consider subjective probabilities is toxic. Frequentist statistics were invented in a (failed) attempt to keep subjectivity out of science in a time before humanity really understood the laws of probability theory. Now we have theorems about how to manage subjective probabilities correctly, and how to factor personal beliefs out from the objective evidence provided by the data, and if you ignore these theorems you'll get in trouble. The frequentist interpretation is broken, and that's why science has p-hacking and a replication crisis even as all the wall-street traders and AI scientists use the Bayesian interpretation. This "let's compromise and agree that everyone's viewpoint is valid" thing is all well and good, but how much worse do things need to get before we say "oops" and start acknowledging the subjective probability interpretation across all fields of science?

The most common stance among scientists and researchers is much more agnostic, along the lines of "use whatever statistical techniques work best at the time, and use frequentist techniques when publishing in journals because that's what everyone's been doing for decades upon decades upon decades, and that's what everyone's expecting."

See also [frequentist_probability], Subjective probability, and Likelihood functions, p-values, and the replication crisis.

Which interpretation is most useful?

Probably the subjective interpretation, because it subsumes the propensity and frequentist interpretations as special cases, while being more flexible than both.

When the frequentist "similar event" class is clear, the subjectivist can take those frequencies (often called [base_rate base rates] in this context) into account. But unlike the frequentist, she can also combine those base rates with other evidence that she's seen, and assign probabilities to one-off events, and make money in prediction markets and/or stock markets (when she knows something that the market doesn't).

When the laws of physics actually do "contain uncertainty", such as when they say that there are multiple different observations you might make next with differing likelihoods (as the [schrodinger_equation Schrodinger equation] often will), a subjectivist can combine her propensity-style uncertainty with her personal uncertainty in order to generate her aggregate subjective probabilities. But unlike a propensity theorist, she's not forced to think that all uncertainty is physical uncertainty: She can act like a propensity theorist with respect to Schrodinger-equation-induced uncertainty, while still believing that her uncertainty about a coin that has already been flipped and slapped against her wrist is in her head, rather than in the coin.

This fully general stance is consistent with the belief that frequentist tools are useful for answering frequentist questions: The fact that you can personally assign probabilities to one-off events (and, e.g., evaluate how good a certain trade is on a prediction market or a stock market) does not mean that tools labeled "Bayesian" are always better than tools labeled "frequentist". Whatever interpretation of "probability" you use, you're encouraged to use whatever statistical tool works best for you at any given time, regardless of what "camp" the tool comes from. Don't let the fact that you think it's possible to assign probabilities to one-off events prevent you from using useful frequentist tools!