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Let's imagine that you still think that it shouldn't matter whether you stay or switch. But you also have become convinced by the observations that it does indeed matter and that you chances of winning are better if you switch rather than stay. You could simply keep separate your two understandings. Alternatively, you could examine what the elements of your old understanding were, and whether by fiddling one or another you can resolve the conflicts between your two understandings.

In using probability, many people make the presumption, consciously or unconsciously, that probability is a property of objects, independent of the observer. Such a presumption might lead one to the conclusion that if there are two closed doors, and one prize, the probability of finding behind each should be the same. An alternate way of thinking about probability was developed in the 1800's by the British mathematician Presbyterian minister Thomas Bayes. Bayes suggested that probability should be thought of not as a independent, fixed characteristic of objects and circumstances in the world but rather as a measure of a given observer's degree of certainty or uncertainty about the outcome of an observation yet to be made. Because of this probability is expected to change depending on the observer's prior experiences.

An interesting way to think about this is the case of the Three Doors Problem is to compare two people, one who has been playing the game (Albert) and knows that one door was open and another (Marie) who arrives on the scene after the first door was opened and is asked without knowledge of that, what the probabilities are of the prize being behind each of the two closed doors. Marie has no prior relevant experiences and so replies entirely appropriately, that there is an equal probability of the prize being behind either of the two closed doors (50%/50%). Albert, on the other hand, has an additional relevant observation: that Serendip (the host) knew where the prize was an opened a door which didn't have the prize behind it. It's this additional observation that distinguishes between Albert as an observer and Marie as an observer and makes the different probabilities they each assign appropriate. It is not the door that has a probability, it is an observer with a particular set of experiences, observering the door(s).

Another way to think about this is to imagine a small modification in the game. If instead of you picking one of the three doors at the outset, Serendip simply opened one of the three doors, showed you the prize wasn't there, and then asked you to pick between the two remaining doors, then the probabilities would indeed be 50%/50%. Again, probability is not a deducible and fixed property of objects in a present circumstance. It depends on what an observer knows about the circumstances and events up to that point. A similar conclusion comes from thinking about situations with more than three doors.

Maybe now your more comfortable that you have reached a broader understanding, i.e. now you are experiencing less conflict between two understandings. Or maybe not; perhaps you need to see the itemization of all possible cases. What is actually different, depending on the circumstances of the game, is the percentage of future cases in which the prize is behind each of the remaining doors. And it is one's knowledge of that variation that leads to the differing probabilities.

Whether you are now more comfortable or not, you are at least on your way to broader understanding of probability. You have recognized a presumption that you have previously made about probability, consciously or unconsciously, and noticed that it may not be a reliable presumption. From now on, you will think about probability in a new way. In recent years, there has come to be broad recognition that Bayes' approach to probability yields more empirically validated probability estimates in a number of cases. To read more about Bayesian probability, visit the resource page here.

Hands on understanding unconscious, intuitive

Experimental understanding conscious, observational

Broader understanding rational, generalizable, unified

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