Rolling dice, an excellent model of probability. Details

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.

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

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,

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

• 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.

Show examples

Hide

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.

Show probability distribution table of two dice being rolled

Hide

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.

Show example

Hide

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

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

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.

• 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.

• Math2.org (more info) has information on statistical distributions in English and Spanish
• Mathworld has extensive definitions from probability and statistics