Introduction to Probability Theory, Part 3

All probabilities are conditional

Ok, now we come to an interesting point. If probability is attached to propositions, and propositions are about objective things that can be true or false, is it right to say that "the objective probability of proposition X is so and so"? What if you and me disagree in our probability assignment about the same proposition - I feel that this page is top-notch and you feel that it is mediocre, without either of us knowing the actual public rating?

Regardless of what you might have been taught about "events" having some inherent "probabilities" that "we" are trying to calculate, the above example of disagreement about probability of a proposition is a perfectly normal situation. We all know that it happens all the time. Just turn on your TV and look at some programme with folks arguing like crazy about different issues. Obviously, independently of how concrete a proposition is, people may disagree on its probability - it is a measure of their degree of confidence, not your degree of confidence after all! Now, the next question naturally is: why can different people have different degrees of confidence in truth of the same thing?

The answer is their different information context. Whether or not you hold some proposition for likely strongly depends on what other propositions you believe in. In a way, all the different propositions are related in our heads, and we are usually quite ready to change our opinion on one proposition after learning something about another. For example, you might be somewhat certain that I'm a native English speaker after reading my text, but if you could throw a brief glance at my passport, it would change your assessment. If you saw an entry "American" under nationality in it, you'd become (almost) certain about the truth of that proposition. On the other hand, if you saw some other nationality, you would become almost certain that the proposition is false. Now, someone else might not have had the same opportunity of looking at my passport and therefore assign a different probability.

Generally, what probability we assign to some proposition depends on what we already know about some other propositions. In mathematical speak, we refer to conditional probability - the probability assigned to X given that we already know that Y is true. In fact, for all practical purposes, all probabilities are conditional. Instead of saying that two different people assign a different probability to the same proposition, we may just as well say that they are just giving us two different probabilities concerning this proposition. The first person is giving us the probability conditional on A (her state of information), the other person is giving the probability conditional on B (her different state of information). There is nothing strange or disturbing about the discrepancy in numbers that arises then. On the contrary, if we could bring the two persons to believe exactly the same set of the "remaining" relevant propositions, they would agree perfectly on the probability assigned to the one uncertain proposition because they would effectively think exactly the same and thus lack any reason to disagree. This convergence of opinions is not easy to achieve, but it is not as far-fetched as it might seem. It can and routinely does happen during practical investigations and in science.

The important point to take from this part is: probabilities are assigned to propositions, but they are not properties of the propositions alone. Instead, a probability is a property of the proposition in question together with all other propositions held to be true by the person who assigned the probability. In fact, we can forget about the person altogether and just represent her by the totality of all propositions she knows to be true.

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