How to study probability

[X89codered89X]

I'm fundamentally confused about this field of mathematics, yet I need to understand it. The field seems to be dominated by set theory notation, and yet, set theory is nowhere to be seen in any of the typical example problems you might see. I fail to find the intuition anywhere, and I feel like I'm looking at the field wrong. Is there some sort of intuition-based textbook or introduction to probability.
Any help

 

[Stephen Tashi]

I'm fundamentally confused about this field of mathematics, yet I need to understand it.

It would be helpfull to your would-be advisors to understand why. What is your objective?

The field seems to be dominated by set theory notation, and yet, set theory is nowhere to be seen in any of the typical example problems you might see.

That's not true in intermediate and advanced texts. In elementary texts, it might be.

I fail to find the intuition anywhere, and I feel like I'm looking at the field wrong. Is there some sort of intuition-based textbook or introduction to probability.
Any help

You should ask more specific questions. People don't know how to correct your wrong way of looking at probability unless you explain what it is.

Unless your goal is very simple, you'll get in trouble trying to take a simple intuition based approach to probability. Such approaches tend to substitute over-simplified and wrong ideas for the mathematics.

 

[chiro]

Hey X89codered89X.

The basic intuition for probability comes from knowing how to classify events in a probability space in a wide variety of contexts and how to use that to answer specific probabilistic questions related to those events.

The Kolmogorov axioms are used to specify the absolute basic conditions mathematically and they deal with sets corresponding to particular "events" in the probability space.

But this context isn't just a univariate situation: it applies to multivariate, joint, conditional, as well as general transformations of the previous categories.

But even with probability you have different contexts. In the one sense the probability space corresponds to a set of events that have a well defined meaning, often one that is tangible and physical like the number of heads given so many tosses, but then you get into mathematics which is invariant to a particular problem or process and this comes when you look at all the identities to do with things like MGF's, CoVariance, Pivotal Quantities and so on.

The connection comes from knowing how the problem at hand relates to all pieces of necessary mathematics and statistics/p-values generated and how these values relate to the context of the problem at hand.

You have on the one hand, all the mathematical machinery including the axioms, transformation theorems, identities, proofs, and so on and on the other hand you all this other stuff that is completely contextual that is hard or impossible to quantify accurately that will affect everything you do mathematically and what kind of conclusions you draw from such results.

But if you starting out, you should think about the basic core of probability which is relating events to some probability space in the form of giving an event a probability.

Giving an event a probability and understand that in the context of the various kinds of distribution functions will help immensely when you relate these with the other mathematical theorems later on.