Geologic context:
forecasting, hazard assessment, error analysis, recurrence interval, radioactive decay

# Understanding odds and probability in the geosciencesby Dr. Eric M. Baer, Geology Program, Highline Community College

Jump down to: Teaching strategies | Materials & Exercises | Student Resources

Probability is the study of the chance that a particular event or series of events will occur. Typically, the chance of an event or series of events will occur is expressed on a scale from 0 (impossible) to 1 (certainty) or as an equivalent percentage from 0 to 100%.

The probability (Pf) of a favorable outcome is
Rolling dice, an excellent model of probability. Details

### Pf = f/n

where:
• n is the total number of possible outcomes and
• f is the number of possible outcomes that match the favorable outcome criteria.
• The analysis of many events governed by probability is statistics, covered elsewhere.

## Teaching Strategies: Ideas from Math Education

### Put quantitative concepts in context

There are a number of geologic contexts in which to introduce and explore probability
Molten lava flowing into the ocean. Details
• Hazard assessment including earthquakes, floods , and volcanic eruptions
• Extinctions and extinction rates
• Error and data analysis including uncertainty and error propagation
• Forecasting

### Use multiple representations

Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. Here are some ways that probabilities can be represented.
• Verbal representation
Probabilities are often given verbally as a percentage, such as a "50% chance." Other times probabilities are even represented by qualitative terms such as "sure," "unlikely," or "almost certain." For example,
The U.S weather service defines several verbal probabilities:
• slight chance - 20% probability
• chance - 30-50 % probability
• likely - 60-70 % probability
• no qualifier - 80-100% probability
• Numerical representation
• One common source of confusion is that probabilities are represented in many differing ways all of which mean the same thing. Probabilities can be represented as a ratio, percentage, fraction or as a decimal; I often point this out to students, so they are alert to the multiple ways we represent odds. This often brings up difficulties and fears with basic mathematics including fractions, percentages and ratios.
Some ways of representing the same probability
ratio - written as 1:3 or 1 in 3.
fraction - 1/3 or sometimes even as an unreduced fraction 3/9
percentage - 33%
decimal - 0.33
• Tabular representation
• Tables are often used to show probabilities across a sample space, called a probability distribution.
Probabilities of a sum given a single roll of two dice
 Sum probability (fractional) probability (percentage) 2 1/36 2.78% 3 2/36 5.56% 4 3/36 8.33% 5 4/36 11.11% 6 5/36 13.89% 7 6/36 16.67% 8 5/36 13.89% 9 4/36 11.11% 10 3/36 8.33% 11 2/36 5.56% 12 1/36 2.78% TOTAL 36/36 100%
• Graphical representation
• There are several graphical ways to portray both probabilities and probability distributions.

Graphical ways of showing probabilities

Bar
Pie chart
Number line

Graphical ways of showing probability distributions

Bar chart
Distribution probability
Cumulative percent

• Model representation
• There are many models one can use as an introduction to probability, including dice rolls, coin tosses, causes of death, and chances of floods.
One example is the probability of death due to individual causes. For instance, a typical American has a one in three chance of dying in a heart attack, a one in 1/100 chance of being killed in auto accident and an estimated 1 in 20,000 chance of being killed by a meteor strike. These probabilities can then be used to stimulate a discussion if geologic hazards, impact of risk perceptions, and mathematical calculations of probability.

Chance/probability of death from various causes for an average American in a given year.
 Cause of death Probability Aids/HIV infection 1 in 19,000 Airplane Crash 1 in 2,736,000 Alcoholic Liver Disease 1 in 22,000 Car Crash 1 in 6357 Earthquake 0 to 1 in 40,000,000 Falls 1 in 20,688 Flood 1 in 6,700,000 Food Poisoning 1 in 53,000 Heart disease 1 in 387 Homicide 1 in 16439 Suicide 1 in 9350
• Algebraic representation
• The probability (Pf) of a favorable outcome is
Pf = f/n
where n is the total number of possible outcomes and f is the number of possible outcomes that match the favorable outcome criteria.

### Use technology appropriately

Students have any number of technological tools that they can use to better understand quantitative concepts – from the calculators in their backpacks to the computers in their dorm rooms. Probability and recurrence intervals can make use of these tools to help the students understand this often difficult concept.

• Graphing calculators
• Graphing calculators are an easy way for all students to enter data and to see what a curve of that data looks like. All graphing calculators are slightly different and students may need help with their particular model. There are some helpful hints for some calculators at Prentice-Hall's Calculator help website .
• Computers
• Probability provides an excellent opening for an introduction to the use of spreadsheet programs. Spreadsheet programs can be used to develop probability distributions and to generate graphs of these distributions. An example to start with might be the probability of rolling a particular sum for 1, 2, 3 and more sets of dice. Students are likely to encounter spreadsheet programs in many of their classes and they are excellent tools for visualizing the shape of an equation.

### Work in groups to do multiple day, in-depth problems

Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions.

Probability is a concept that comes up over and over in introductory geoscience: Volcanic eruptions, mass extinctions, earthquakes, floods, confidence intervals, etc. When each new topic is introduced, make sure to point out that they have seen this type of mathematics before and should recognize it.

## Teaching Materials and Exercises

• Determining Earthquake Probability and Recurrence
• In this exercise students gather real web-based data on earthquakes in the Pacific Northwest, calculate recurrence intervals on these events and make estimations of recurrence intervals on future events. Includes analysis of error and societal implication discussions. Designed as a combination in-class and web-based homework assignment. This could easily be altered to look at earthquake possibilities for any geographic area.
• Is there a present volcanic hazard?
• The volcanic processes that created the Yellowstone Geoecosystem are still active. One way that geoscientists determine the likelihood of a certain type of eruption occurring is by documenting how often they have occurred in the past (i.e. the recurrence interval). We will explore the types of eruptions that Yellowstone is likely to produce and the probability of their occurrence. In this self-guided exercise, we will explore only one of a number of possible pathways to answer the question.
• Chance news
• This site contains materials to help teach a Chance course. Chance is a quantitative literacy course. It includes "The Chance News" a newsletter that includes lesson plans and articles with discussion questions from the Chance magazine. A source for supplementary material for teaching or introducing basic probability.
• Texas instruments activity page
• has several activities in probability and statistics which use TI calculators. Click on "Activity Exchange" and then select "Math" on the left sidebar under "Browse by subject." Then select "Probability & statistics" and "Probability" for a large selection of activities. Many are listed as being for grades 9-12, but they would be relevant for an introductory level geoscience course.
• The Math Forum
• has extensive links to classroom resources. Look in Exploring Data for finding and displaying data sets. Also included are an Internet Mathematics Library, Course Materials & Lesson Plans, Collections of Course Materials & Lesson Plans, Problems and Puzzles, Probability and statistics software, internet projects, and Reference Materials.

## Student resources

• MathHelp has general information on probability and probability distributions designed for students
• Math2.org has information on statistical distributions in English and Spanish
• Mathworld has extensive definitions from probability and statistics