Geologic context:
forecasting, hazard assessment, error analysis, recurrence interval, radioactive decay
Understanding odds and probability in the geosciences
by Dr. Eric M. Baer, Geology Program, Highline Community College
Jump down to: Teaching strategies | Materials & Exercises | Student ResourcesProbability 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.
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.
Numerical representation
Tabular representation
Graphical representation
Bar
Pie chart
Number line
Graphical ways of showing probability distributions
Bar chart
Distribution probability
Cumulative percent
Model representation
Algebraic representation
- 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,
Show U.S. weather service verbal probabilities
Hide 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
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
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% |
There are several graphical ways to portray both probabilities and probability distributions.
Graphical ways of showing probabilities
Graphical ways of showing probability distributions
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.
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 |
The probability (Pf) of a favorable outcome is
- Pf = f/n
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
- Computers
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 (more info) .
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.
