# Probability's Connection to Calculus

If it is the case that I wish to concentrate my efforts in learning Probability, Statistics and Game Theory, then is it also the case that I must become proficient in calculus?

Is calculus a foundation for all mathematical learning... or are there other foundations that would better pertain to Probability, Statistics or Game Theory.

-When reading books such as "Introduction to Probability" "Introduction to Statistics" or "Introduction to Game Theory" I always end up encountering terminology or notation that I cannot understand and therefore cannot progress through the rest of the book. I am unfamiliar with the terminology/notation and my immediate reaction is that these terminology/notation are related to calculus in some way ---> am I mistaken in this assumption? Are there mathematical foundations outside of calculus?

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Probability is a very broad subject. Can you be more precise? Can you, for example, list some of the terminology and notation that you don't understand? Have you tried looking them up on Wikipedia? – Qiaochu Yuan Jul 19 '11 at 2:25
I have looked most of them up on Wikipedia. The biggest problems that I have been facing were associated with set theory. Almost every time set theory came up infinite sets followed and the summations started to pop up (as if it was supposed to be assumed that I understood their meaning) this is what caused me to stop reading literature on probability or game theory and focus just on calculus. Which brings me to the question of whether this was the right choice. I'm unfamiliar with how to represent symbols on this sight so it's hard to give examples now. – Derek Jul 19 '11 at 2:41
Even the use of subscripts and superscripts began to confuse me. I've searched for an introduction to set theory, but it seems to cause more problems because there's no basic explanation for the use of any of the notation... this might have to do with my lack of experience with higher level math, but it's hard to find a bridge. I've tried to google and wikipedia the notation for the topics, but each explanation for these notation or terminology seems to require more assumed knowledge. – Derek Jul 19 '11 at 2:46
Well, it is quite important to understand infinite series in probability, and generally the first place you learn about infinite series is in a calculus course. Are you a high school student? Is there a school nearby where you could take higher-level courses? – Qiaochu Yuan Jul 19 '11 at 2:47
Thank you very much for the help. I really needed the confidence to commit to learning calculus and I have definitely obtained that. – Derek Jul 19 '11 at 3:04

Well, calculus is not a tool for intuition, it's name says it : it's meant to calculate. I believe the purpose for calculus is that when our intuition fails to explain things, we rely on theorems and then use calculus to confirm our intuition. We don't use calculus as a fundamental : it's a tool. The fundamentals lie behind the ideas of those theories. For instance, probability without calculus is just a bunch of nice ideas, because there's always series in there, integrals, and such, so you need to calculate stuff if you want to be able to say something interesting...

EDIT : As an example, so that downvote can be removed...

Guessing the expectation of a continuous random variable can be a pain if the variable has a non-trivial expression, and probabilistic theorems might not make it easier to solve it : using calculus to compute the integral might make it work. Here, calculus helped where probabilistic intuition couldn't ; that doesn't mean there is no intuition in calculus. That is what I meant by "Calculus is not a tool to develop intuition. It's a tool for computing."

All this was said in a probabilistic context. Calculus lovers, I am not offending you : calculus is amazing on its own, too.

Hope that helps,

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That was awesome. That did help a lot. I'm more excited and motivated to learn calculus now that my mind has settled on it's necessity. – Derek Jul 19 '11 at 2:26
"We don't use calculus as a fundamental : it's a tool. The fundamentals lie behind the ideas of those theories." This makes sense now. – Derek Jul 19 '11 at 2:27
@Derek: you should show your appreciation of an answer you like by upvoting and, if you like it enough, accepting the answer. It makes us feel good. – mixedmath Jul 19 '11 at 2:29
Haha. Gotcha. Will do. I need more reputation... will definitely remember that though. – Derek Jul 19 '11 at 2:30
A great deal of intuition accompanies Calculus. That is it least as powerful as its ability to compute exact areas, volumes and other quantities. It is an acquired intuition that is extremely valuable. I find myself 100% at issue with this statement. – ncmathsadist Jul 19 '11 at 2:55

You're asking several related questions here, and some of them I'm not sure you're asking, but here's one I'm pretty sure you're asking: no, calculus is not the only foundational thing you need to learn in mathematics. Even very applied mathematicians should learn lots of linear algebra as it is quite ubiquitous, and it is always a good idea to know a little analysis (roughly, rigorous calculus) and topology. Depending on what you're interested in, it's also a good idea to learn some abstract algebra, maybe a little number theory, and some more analysis and topology. And depending on what kind of probability you're interested in, it would be a good idea to learn some combinatorics as well.

This is not a complete list, exactly, but it's a start. There's a lot of mathematics out there.

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