Most problems students encounter in algebra and calculus have a definite answer -- the ball will go 20 feet high; the cars will collide in 30 seconds. But there are many situations in real life that don't have definite answers -- Will it rain tomorrow? What number will come up on the die when you roll it? Probability is the study of these types of uncertain situations, where we cannot say definitely what will happen, yet can say that over the long run, a certain pattern will occur (for example, if you toss the die many times, each number will come up about the same proportion of the time.) Since these types of uncertain situations occur in so many areas, probability is important in a wide variety of fields, including weather, medicine, political analysis, business, environmental risks, semichip manufacturing, quantum mechanics, sports, insurance, traffic flow, and lotteries.

Probability, like most mathematics courses you will take at the University, isn't about manipulating formulas (although you will need to do some of that, too). Probability is mostly about thinking. The goal is to learn to think logically about the incomplete information we have about the real world.

For more about probability and its history, click here.

Standards II (Patterns and Algebra), IV (Probability and Statistics), and V (Mathematical Processes) of the SBEC Secondary Math Standards include the following standards relevant to this course:

"The beginning teacher of mathematics is able to:

• (2.17s) work with patterns with random variations (p. 7) …
• (4.5s) use the concepts and principles of probability to describe the outcome of simple and compound events;
• (4.6s) explore concepts of probability through data collection, experiments, and simulations; …
• (4.8s) use the graph of the normal distribution as a basis for making inferences about a population….
• (4.12s) calculate and interpret percentiles and quartiles; …
• (4.15s) determine probability by constructing sample spaces to model situations; and
• (4.16s) make inferences about a population using the binomial and geometric distributions. …
• (4.21s) calculate probabilities using the axioms of probability and related theorems and concepts such as the addition rule, multiplication rule, conditional probability, and independence;
• (4.22s) apply concepts and properties of discrete and continuous random variables to model and solve a variety of problems involving probability and probability distributions; …
• (4.27s) use the law of large numbers and the central limit theorem …(pp. 13 - 16)
• (5.1s) apply correct mathematical reasoning to derive valid conclusions from a set of premises; …
• (5.3s) use formal and informal reasoning to explore, investigate, and justify mathematical ideas;
• (5.4s) recognize examples of fallacious reasoning;
• (5.5s) evaluate mathematical arguments and proofs;
• (5.6s) provide convincing arguments or proofs for mathematical theorems. (p.17) …
• (5.7s) recognize that a mathematical problem can be solved in a variety of ways, evaluate the appropriateness of various strategies, and select an appropriate strategy for a given problem;
• (5.8s) evaluate the reasonableness of a solution to a given problem;
• (5.9s) use physical and numerical models to represent a given problem or mathematical procedure;
• (5.10s) recognize that assumptions are made when solving problems and identify and evaluate those assumptions;
• (5.11s) investigate and explore problems that have multiple solutions;
• (5.12s) apply content knowledge to develop a mathematical model of a real-world situation and analyze and evaluate how well the model represents the situation;
• (5.13s) develop and use simulations as tool to model and solve problems; …(p. 18)
• (5.15s) explore problems using verbal, graphical, numerical, physical, and algebraic representations.
• (5.16s) recognize and use multiple representations of a mathematical concept;
• (5.17s) apply mathematical methods to analyze practical situations;
• (5.18) use mathematics to model and solve problems in other disciplines, such as art, music, science, social science, and business. … (p. 19)
• (5.21s) translate mathematical statements among developmentally appropriate language, standard English, mathematical language, and symbolic mathematics; …
• (5.23s) use visual media such as graphs, tables, diagrams, … to communicate mathematical information;
• (5.24s) use the language of mathematics as a precise means of expressing mathematical ideas." (p. 20 )

How does this course relate to the Texas Essential Elements (TEKS) for secondary mathematics?

• Probability topics are listed in each grade level K - 8. (Click here to see a list of probability and statistics topics included in grades K - 8. Click here for links to examples of probability and statistics activities in grades K - 8.)
• The high school course Mathematical Models with Applications includes a probability component:

• Probability, with emphasis on description of data distributions, is one of the four units of AP Statistics. Enrollment in AP Statistics has been growing steadily recently, so there is a need for teachers prepared to teach it. (Click here to go to the College Board AP Statistics home page. Click here to see a probability lesson from an AP statistics course at Rosemead High School, in Rosemead, CA.)

The Principles and Standards include Data Analysis and Probability as one of their five content strands running through all grade levels.

• Click here to see the Data Analysis and Probability overview.
• Click here to see the Data Analysis and Probability Standard for Grades Pre-K–2.
• Click here to see the Data Analysis and Probability Standard for Grades 3 - 5.
• Click here to see the Data Analysis and Probability Standard for Grades 6 - 8.
• Click here to see the Data Analysis and Probability Standard for Grades 9 - 12.

As probability has become more important in the modern world, it has been included increasingly in school curricula. Unfortunately, many teachers leave out the probability topics, often because they are not familiar with the subject themselves. So it is important for future math teachers to have a good background in probability, so that they don't shortchange their students.

How does this course relate to other courses I will be taking?

• Calculus (especially improper integrals, series, and double integrals ) is used in M 362K..
• The set theory and combinatorics in M 325K (Discrete Mathematics) are helpful in M 362K, so it is recommended that you take M 325K before M 362K. PHL 313K can also help give preparation in set theory, logic, and functions.
• M 362K is a prerequisite for M 358K (Applied Statistics) and is used extensively in that course.
• You may find probability and statistics mentioned in your UTeach course Research Methods
• You may decide to include probability topics in your term project in the UTeach course Project Based Instruction.
• If you are working toward the BS-Teaching Option math degree, you will be required to take a depth course. One option (M 378K: Mathematical Statistics) has M 362K as a prerequisite and uses material from this course extensively.

How can I get the most out of this course?

• Be sure to learn your calculus thoroughly and review it as needed in M 362K.
• Take M 325K before you take M 362K.
• Expect to look at the world from a different perspective in M 362K than you might be used to.
• Be sure to learn technical vocabulary and not confuse it with everyday usage. For example, the words "random" and "normal" are used in probability, but with more precise meanings than in everyday usage.
• Pay attention to the items in SBEC standard V (listed above) as you take this course. Practicing setting up problems (i.e., modeling) is an important part of learning probability.
• Learn for understanding and retention, since you will be using concepts from M 362K in M 358K and later in your teaching.

Here are some:

• A dice rolling simulation, showing the idea of a probability distribution.
• Let's Make a Deal applet.
• Cut-the-Knot's Probability page, with links to some probability problems.
• A Fathom document to simulate the following problem: A randomly shufffled deck of 52 cards is dealt into 26 pairs. How many of the pairs make a "21" hand?
• A Central Limit Theorem illustration.
• Probability and Eyewitness Accounts