# How hard is Probability?

1. Mar 15, 2012

### physicsdreams

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

2. Mar 15, 2012

### Jorriss

Proof based probability, quite complicated.

Computational probability, still can be pretty damn complicated. If you are taking a probability sequence that is lower division (presumably not calculus based then) then of course it can be made very simple and is often designed for non scientist/engineer/math majors.

3. Mar 15, 2012

### Broccoli21

The courses at my school at least are:
1) Intro to prob/stat (calc based)
2) Intermediate probability (heavy proofs, lots of calculus/analysis)
3) Probability Theory (measure theory, analysis, etc.)

Probability can indeed get pretty hard. I found upper-level probability courses probably as hard as my real analysis ones. Its only the intro classes (computational, not even calc based sometimes) that are really easy.

4. Mar 15, 2012

### chiro

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.

5. Mar 15, 2012

### twofish-quant

It's because most basic undergraduate probability and statistics courses are geared to teach the simple parts of probability and leave out the complicated bits.

6. Mar 16, 2012

### 20Tauri

It depends on what kind of probability class you're talking about. Probability theory/mathematical statistics at my college is taught as an advanced sequence and requires Calc III and Linear Algebra as prereqs. I just took the first course, and I'd say it's less abstract than Linear Algebra but not necessarily easier, if that makes sense. We did some proofs, but it was mainly computational, with a lot of integrals and stuff like that.