physicsdreams said:
It seems like everyone I know believes that probability is much easier than Calc, but I believe that's a subjective point of view.
I know that Calculus has various levels (i.e. I,II,III), but is it really all that much more difficult than probability?
How many different 'levels' of probability are there, and can't probability be just as hard, if not harder than Calculus?
Thanks
Hey physicsdreams and welcome to the forums.
The concept of probability is pretty simple, but when you have to apply it or work things out when it's not just given in an equation then it takes practice before you start building up the intuition.
In terms of how hard it can get, I would compare the situation of calculus in your first year with analysis in graduate school. In probability you can end up taking probability and then looking at it in a measure theoretic context just like you do in an analysis course.
Just like with pure mathematics, you can get results that are a lot more abstract. In Hilbert spaces we consider systems that have an infinite-number of dimensions. What if you want the sum of infinitely many probability distributions? How do you deal with convergence?
But it's a good idea to know what probability means intuitively before applying all the analysis, measure theory and so on. This means doing probability problems in the context of normal calculations, conditional probability calculations, simulation, normal models (binomial, poisson, exponential, normal, etc) markovian modelling, and stochastic processes.
In addition it would help if you also did a statistics component to help put a lot of this into a different but very recommended perspective.
The real thing about probability is knowing a) what the atomic events are, b) what the assumptions are and c) what these do to define the probabilistic functions for the process.
If you don't know how to do the above and don't understand how this helps you calculate probabilities then you'll be stuck no matter what level you are at.
Also one final thing is to rely on the mathematics over intuition at least initially. If you try and use your intuition, then you'll probably end up being wrong. Build up your intuition from your experience solving problem, not the other way around.
In terms of the 'levels', different perspectives will focus on different models. Markovian models deal with the fact the only thing that affects the probability of the next event is the event before that. This assumption is used in so many fields that it has become a field in itself which is part of what is known as "Applied Probability". It has its own theoretical results concerning how to find long-term distributions, means and so on.
The basic thing is that you need to understand how each 'kind' of probability is different from another. In mathematics we don't just don't start out by studying the most general way (and we don't end up that way either): rather we look at a model that is useful or widely used and then we make small increments of progress on understanding that particular model or framework.