# Bloomington Tutors Blog Finite math, K201, calculus, and statistics explained

## Going steady (state) with Markov processes

Finite M118

After the finite midterm, you may have been confused and annoyed when the class seemed to abruptly shift from probabilities and permutations to matrices and systems of equations. They seem to have nothing to do with each other!

Well, today is your lucky day. Probability theory and matrices have finally met, fallen in love, and had a beautiful baby named Markov.

Let me explain. Markov processes are widely used in economics, chemistry, biology and just about every other field to model systems that can be in one or more states with certain probabilities. These probabilities change over time.

For example, we might want model the probability of whether or not Amazon.com turns a profit each quarter. So, Amazon.com (the system) can be in two possible states at the end of each quarter: either they turn a profit, or they don't. Their chance of turning a profit this quarter depends on whether or not they turned a profit last quarter. For example, if they profited last quarte...

## How to understand and solve Leontief input-output model (technology matrix) problems

Finite M118

Today, let's take a look at everyone's favorite matrix application problem, Leontief input-output models. You might know them simply as "technology matrix" problems, but actually the technology matrix is only one part of the problem. The really interesting part is in the derivation of the matrix equation - something that most finite math courses seem to gloss over in the end-of-semester frenzy.

So, let's take a look at a typical "technology matrix" problem, and see if we can't understand how the problem actually works. Hopefully, it will make it easier to remember than an arbitrary formula.

Example: An economy consists of two codependent industries - steel and lumber. It takes 0.1 units of steel and 0.5 units of lumber to make each unit of steel. It takes 0.2 units of steel and 0.0 units of lumber to make each unit of lumber. The economy will export 16 units of steel and 8 units of lumber next month. How many units of steel and lumber will they need to product...

## When to use the z-test versus t-test

Statistics

How do I know when to use the t-test instead of the z-test?

Just about every statistics student I've ever tutored has asked me this question at some point. When I first started tutoring I'd explain that it depends on the problem, and start rambling on about the central limit theorem until their eyes glazed over. Then I realized, it's easier to understand if I just make a flowchart. So, here it is!

Basically, it depends on four things:

1. Whether we are working with a mean (for example, "37 students") or a proportion (e.g., "15% of all students").
2. Whether or not we know the population standard deviation ($\sigma$). In real life we usually don't, but statistics courses like to contrive problems where we do.
3. Whether or not the population is normally distributed. This is mainly important when dealing with small sample sizes.
4. The size of our sample. The magic number is usually 30 - below that is considered a "small" sample, and 30 or above is consi...

## Practice for your finite exams with our sample problems!

Finite M118

Welcome, new and returning IU students! I hope everyone had a great summer and is ready (sorta?) to get back to class. Here at Bloomington Tutors, we've been working hard all summer and getting ready to help you succeed.

We've launched our practice exam system for finite math (M118), and hope to start adding materials for calculus and other courses soon as well. Check out our finite questions here: https://bloomingtontutors.com/quiz/finite-math

They're interactive multiple-choice questions, with explanations for the wrong answers (i.e., we try to guess the common mistakes you might have made to get that answer, and offer you an explanation).

Enjoy!

## A linear programming word problem - with a surprise twist!

Finite M118

Today, I thought I'd tackle a problem from one of the most "popular" courses (read "required for almost all majors") at this school - finite math. For those of you who haven't heard of this course before, finite math is actually an amalgam of miscellaneous topics from probability theory, linear algebra, and stats, created by math departments at various universities as a way to introduce college freshmen to rigorous, analytical thinking. It's not a real field of mathematics, in the sense that you won't find any professional mathematicians who specialize in "finite mathematics." However, there are mathematicians who specialize in the topics introduced in this course, such as combinatorics and probability.

One of the topics covered in finite math ("finite", by those in the know) is linear programming. This is basically a fancy term for a constrained optimization problem consisting of linear constraints and a linear objective function. In this blog post, I will tackle the followin...