Is probability determined by perspective?
The other answers have adequately explained this variation of the Monty Hall problem, so I'm going to focus on the question:
My question: is probability determined by perspective?
The answer is: It depends.
Broadly speaking, there are two schools of thought about what a "probability" is and how it should be used. The frequentist believes that probability describes something like a roulette wheel, which can be spun many times but always behaves the same (i.e. it always produces the same distribution of results). When we say $P(X) = \frac{1}{6}$, we mean that, if a given event is tried many times, in the long run the outcome $X$ will happen one sixth of the time. This convergence is guaranteed by the law of large numbers. Probability is thus an objective fact about the universe, and not something subject to a person's perspective. In this worldview, the probabilities in the Monty Hall problem are always 1/3 and 2/3, regardless of whether we know which is which.
The Bayesian, on the other hand, sees probability as a degree of belief. You might think of this like in a court of law: In order to convict the defendant of some crime, we need to be 99% certain that the defendant committed the crime. Seeing incriminating evidence may raise our subjective belief in the defendant's guilt, while exculpatory evidence would lower it, both according to Bayes' theorem. When we're making a verdict, we ask ourselves whether the defendant has at least a 99% probability of having committed the crime. In the frequentist worldview, this is a nonsensical question; either the defendant is truly guilty or the defendant is truly innocent, and the probability is accordingly either 100% or 0% (we just don't know which). Similarly, in the Bayesian worldview, the Monty Hall problem is nonsensical unless you specify the person whose worldview we are following and their subjective prior probabilities for each door. Bayesian reasoning, then, could give you a 50-50 split for your hypothetical second contestant, but only if she started with 33-33-33 priors, and then only if there is no other evidence allowing her to distinguish between the two remaining doors.
It is also important to recognize that the frequentist and Bayesian approaches are not mathematically distinct as both probabilities are subject to the same mathematics (i.e. each system admits both the law of large numbers and Bayes' theorem). What differs is how the math is applied to the real world. Because the frequentist deals in objective probability, they cannot tell you "the probability that candidate X wins the election."[1] Because the Bayesian deals in subjective probability, they cannot tell you much of anything without a set of prior probabilities,[2] which are necessarily tied to a particular observer at a particular point in time and space. Ultimately, both systems inevitably require making certain assumptions about how your data relates to the real world. So you should examine those assumptions with care before blindly accepting the result which they have produced.
[1]: The election happens only once; it doesn't make sense to ask how often candidate X wins. Imagining the election being re-held many times doesn't work either because elections are deterministic, so the same people will vote or not-vote in the same ways every time, and you will get the same result. Instead, you have to engage in a far more roundabout investigation of the likely level of errors in the polls, which gives a less obviously meaningful value as its final output.
[2]: In cases like Monty Hall, some set of priors is typically "obviously correct" (e.g. "All three doors are equally likely to conceal the car"). However, this still has to be explicitly stated as an assumption of the Bayesian method. Many circumstances, including elections, have no obviously correct set of priors (though betting markets may be a good first step). In cases like the court of law, it may be desirable to begin with a set of priors which is "obviously wrong" (we must assume the defendant is probably innocent, even though most criminal defendants are probably guilty).
The probabilities are Door 1: $1/3$ and Door 2: $2/3$ for both contestants, it's simply that one of them knows which doors bears $1/3$ and the other doesn't.
One thing that will make things clearer is the following:
Assume that the contestant who arrives late is told what happened before, then asked what are the probabilities for each door. The correct answer is "one of them bears $1/3$, the other one $2/3$, but I don't know which is which.
Now assume that the second contestant is only told that one door has a prize and the other nothing. He's told nothing more about what happened or how it was decided what door would be the right one. He is then asked what are the probabilities for each door. The correct answer is I have absolutely no clue.
There is a fundamental difference between "I don't know" and "$50/50$". It could very well be that the organisers always put the car behind Door 2, in which case the probabilities would be $0$ and $1$. You can't make up probabilities if you don't know the protocol.
Edit: one last point is that there is a difference between the probability that the car is behind Door 2, and the probability that the second contestant will get the car. If the second contestant knows nothing, and picks a door by flipping a fair coin, then indeed they will get the car with probability $0.5$. This doesn't depend on the probability for each door to be the correct one.
This is an ongoing point of philosophical debate, generally subsumed under the name "interpretations of probability". Joseph Butler famously wrote the following in the introduction to his The analogy of religion, natural and revealed, to the constitution and course of nature (1736). Here, he frames probability as essentially a tool for beings in the world dealing with limited and possibly flawed perspective:
Probable Evidence, in its very nature, affords but an imperfect kind of Information; and is to be considered as relative only to Beings of limited Capacities. For nothing which is the possible object of Knowledge, whether past, present, or future, can be probable to an infinite Intelligence; since it cannot but be discerned absolutely as it is in itself, certainly true, or certainly false: But to us, Probability is the very Guide to Life.
Wikipedia provides the following summary of current interpretations. Note in particular that the "Subjective" interpretation states that probability is fundamentally about "Degree of belief", while the others do not:
- Wikipedia: Probability Interpretations
- Stanford Encyclopedia of Philosophy: Interpretations of Probability