# Math 118: Finite Mathematics

### Title

Finite Mathematics

(3:3:0)

Fall, Winter

Math 110

### Description

Description: This course studies the basic elements and applications of finite mathematics. The first half of the class covers the language of set theory, principles of counting and combinatorics, probability theory for equally likely outcomes, elementary stochastic processes, conditional probabilities, and repeated experiments. The concept of a random variable is developed, along with expectation and variance. The second half of the class explores systems of linear equations and matrix algebra, linear programming, and Markov chains. This course considers a broad range of applications in business, the life sciences, and the social sciences.

## Desired Learning Outcomes

1. Part 1: Probability Models (20 lectures, approx.)
• Sets Theory (3 lectures)
• Describe sets using set-builder notation.
• Solve problems involving set membership, subsets, intersections, unions, and complements of sets.
• Be able to identify the various partitions of a Venn diagram.
• Determine the number of elements in a partition, based on set counting rules for unions, complements, and products.
• Combinatorics & Counting (6 lectures)
• Describe the sample space of an experiment and the set of all possible outcomes using trees and the multiplicative principle, as appropriate.
• Explain what a permutation is, and how many permutations there are for a given set.
• Demonstrate how to count more sophisticated permutation problems involving products and restrictions to sets.
• Use the notion of partitions to reduce permutation problems to combinations, where the order does not matter.
• Be able to solve various hybrid counting problems with and without replacement, with and without order.
• Be able to apply ideas from combinatorics and counting, and formulate real-world problems.
• Probability (8 lectures)
• Be able to describe the notions of outcomes and events in probability, and the axioms of a probability space.
• Use ideas from combinatorics to determine the probabilities of various events, with equally likely outcomes.
• Demonstrate understanding of probabilistic independence.
• Describe a stochastic process, and be able to compute probabilities of events on trees.
• Explain Bayes rule and demonstrate proficiency with conditional probability.
• Be able to use Bayes formula to compute probabilities of various conditionals.
• Be proficient with Bernoulli trials, be able to solve basic problems.
• Be able to apply ideas from probability, and formulate real-world problems.
• Random Variables, Expected Values, Variance (3 lectures)
• Demonstrate understanding of what a random variable is. Describe the probability density function and the distribution of a given random variable.
• Show how to compute the expectation, variance, and standard deviation of a random variable.
• Be able to read a table for a normal distribution and compute the probabilities of given events.
• Be able to apply ideas of uncertainty, and formulate real-world problems.
2. Part 2: Linear Models (20 lectures, approx.)
• Systems of Linear Equations (3 lectures)
• Use elimination and substitution methods to solve linear systems of two or three variables.
• Reduce a linear system into row echelon form, then solve with back substitution.
• Solve a linear system by transforming it into reduced row echelon form.
• Identify whether a system of linear equations has no solution, exactly one solution, or infinitely many solutions.
• Matrix Algebra and Applications (3 lectures)
• Perform matrix algebra operations: addition, multiplication.
• Compute the inverse of a matrix, identify when a matrix is not invertible.
• Study in detail at least one application involving matrix algebra (e.g., Leontief economic models).
• Linear Programming (8 lectures)
• Formulate linear programming problems from various application areas, such as business, resource management, etc.
• Describe the constraints, the feasible set, and the objective function of a given linear optimization problem.
• Solve a linear program using the graphical method, when possible.
• Explain the standard form of a linear program.
• Describe the following concepts from the simplex method: slack variable, pivot column, tableau.
• Be able to solve, by hand, a given linear program using the simplex method.
• Be able to solve, by computer, a given linear program.
• Explain and use the dual formulation of a given linear program.
• Markov Chains (6 lectures)
• Be able to describe a Markov chain.
• Identify when a Markov chain is regular, irregular, and absorbing.
• Describe how to determine the stable probabilities of a regular Markov chain.
• Describe how to compute the fundamental matrix of an absorbing Markov chain.
• Be able to apply ideas from Markov chains, and formulate real-world problems. Determine the transition matrix and states of a given application.

Math 110

### Textbooks

Possible textbooks for this course include (but are not limited to):

Finite Mathematics, 5th edition, Daniel P. Maki and Maynard Thompson, McGraw-Hill 2005