Is probability determined by perspective? My question: is probability determined by perspective?  

The scenario that raised the question for me: 

Initial condition: The Monty Hall problem.  We know the contestant’s original choice of door #1 (of 3 total) is only correct 33% of the time.  After Monty Hall reveals door #3 is incorrect the contestant is asked if he would like to switch his answer to door #2.  We know he should choose to change his answer to the other unopened door (#2) which has a 66% chance of being correct.  He does so.  

However, let’s say once he decided to switch to door #2, and before either door is revealed, another contestant enters the room.  She does not have any knowledge of what has just transpired on stage.  She is offered a choice to pick which door the car behind of the 2 remains closed doors and also randomly chooses door #2.  

Are the probabilities of being correct different for each contestant?  Seemingly they are.  Contestant 1 had 3 doors to choose from, giving him a probability of 33% that door #1 is the answer.  Contestant 2 only had 2 doors to choose from, giving her a probability of 50% after choosing the same door contestant #1 choose.  

If we repeat the experiment 1000 times, what will the numbers turn out to be for door #1?  333 or 500?
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).
Is probability determined by perspective?


The answer is undoubtedly yes, but I don't think the Monty Hall problem illustrates the possibility of different perspectives well. Rather I like Bertrand's Paradox.


  Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?   


Bertrand then gave three answers depending on different probability distributions. The whole point was that for the problem to have a particular solution then the probability distribution for the chord length distribution must be specified in the problem statement since different distributions lead to different answers.