The Course:
MTH321-322: Probability and Statistics

A man and a woman (not related) each have three children. The man’s oldest child is a girl. The woman has at least one girl. Compared to the woman, is the man more, less or equally likely to have exactly two girls?

Dr. Jeff Bay, associate professor of statistics, found this brainteaser in an “Ask Marilyn” column of the Parade magazine a few years ago and has since posed it to students enrolled in his MTH321: Probability and Statistics I class.

Bay says that students, of course, first point out that in real life, most people would just ask the parties involved: “How many girls do you have?” But the professor is quick to remind them that by tackling such puzzlers, they’re building up their problem-solving skills. With more practice and greater understanding of probability and statistics, they’ll be able to answer much harder – and more important – questions. Questions such as the effectiveness of a new pharmaceutical tested in a drug study. Or queries of product safety that come from data collected on an assembly line.

Bay doesn’t have to work hard to make the case for the relevance of probability and statistics. In addition to working real-world problems out of the textbook, many students have been introduced, at least, to the concepts of probability and statistics in other courses, he says.

Making inferences about populations based on samples is not limited to researchers in the natural, behavioral and social sciences. In fact, Bay argues that liberal arts students would be well served to gain a deeper understanding of probability and statistics and, if their schedules allow it, sign up for these two classes. (MTH321, taught in the fall, focuses on probability and includes lessons on counting methods, discrete and continuous probability distributions and their properties, and sampling distributions. MTH322, offered in the spring, deals more with applied statistics and introduces the concepts of point estimation, confidence intervals, tests of hypotheses, and regression.)

“We’re naturally curious, and we often ask ‘What’s the chance of this happening?’ or ‘What are the odds of that?’” he says. “That’s often subjective probability, but we need to understand objective probability to better understand the world around us.

“The liberal arts strives to produce well-rounded and informed citizens who are capable of making good decisions,” Bay continues. “These courses fit a liberal-arts curriculum because an understanding of probability helps us make better decisions.”

Printed in his syllabus for MTH321 are five general and five specific goals for the course. Among them is to “realize that the mathematical sciences are universal languages used by scientists from all areas to seek solutions to real-world problems; that is, see the connection between the mathematical sciences and problem-solving of other disciplines.”

Developed over 300 years ago to understand games of chance, probability looks at predicting what will happen in a specific case (or, in statistical terminology, a sample) based on general premises (that is, a model).

The discovery of this science “caused a shift in people’s paradigms,” Bay says, explaining that before, people may have simply observed what happened. Probability put some rhyme and reason behind the likelihood of things happening, and thus allowed people to make predictions.

The field of statistics began to take off in the mid-1900s with the advent of the Green Revolution, which saw major advancements in agriculture. Creating and testing new fertilizers and crop hybrids, researchers had to find a way to accurately and efficiently make assumptions about their products. Using samples and employing probability models to those samples, they were able to make inferences about the whole.

Recently, statistics has experienced huge growth. Technology has enabled the collection and compilation of an overwhelming amount of data these days, and jobs abound for those who can analyze it accurately and efficiently.

It’s not easy stuff, Bay admits. Students have to devote time to homework and special projects that they propose. Probability and statistical problems often require thinking about the question for a while and applying various models. Sometimes, students have to walk away from very challenging problems to let the brain process all the information. He tells students to look at a probability problem from different angles, thus building those problem-solving skills.

“In statistics, I can often list the steps – the recipe – for getting to a final number. For basic probability, it’s not that easy,” Bay explains. “There isn’t one recipe that you can follow.”

Coming up with the answer to the “Ask Marilyn” question, for example, requires writing out the various combinations of girls and boys. Before putting pen to paper, most students guess the probability of two girls per household is the same. They are wrong.

While explaining the answer, Bay says he takes the opportunity to point out to students that human intuition and how people process information aren’t always very accurate.

“There is just a slight difference, but the probability is greater that the man has two girls than the woman,” he says. “Having that extra bit of information about him [that his oldest child was a girl] makes a difference.”