Also see our main page on What is mathematics, really? Is it made by humans or a feature of the universe? Math in art & poetry.

Probability and statistics

Probability is the measure of the likelihood that an event will occur.

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. Populations can be diverse topics such as “all people living in a country” or “every atom composing a crystal”. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments. – Wikipedia

## Black Swan events

The black swan theory is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight.

The term is based on an ancient saying which presumed black swans did not exist, but the saying was rewritten after black swans were discovered in the wild. The theory was developed by Nassim Nicholas Taleb to explain:

The disproportionate role of high-profile, hard-to-predict, and rare events that are beyond the realm of normal expectations in history, science, finance, and technology.

The non-computability of the probability of the consequential rare events using scientific methods (owing to the very nature of small probabilities).

The psychological biases which blind people, both individually and collectively, to uncertainty and to a rare event’s massive role in historical affairs.

- Wikipedia

See our article The Black Swan, by Nassim Taleb

## Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

High School: Overview of Science and Engineering Practices

By the end of high school, students should have an understanding of and ability to apply each science and engineering practice to understand the world around them. Students should have had many opportunities to immerse themselves in the practices and to explore why they are central to the applications of science and engineering. Some examples of these science and engineering practices include:

1. Define a design problem that involves the development of a process or system with interacting components and criteria and constraints that may include social, technical, and/or environmental considerations.

2. Develop and/or use a model (including mathematical and computational) to generate data to support explanations, predict phenomena, analyze systems, and/or solve problems.

3. Plan and conduct an investigation, including deciding on the types, amount, and accuracy of data needed to produce reliable measurements, and consider limitations on the precision of the data.

4. Apply concepts of statistics and probability (including determining function fits to data, slope, intercept, and correlation coefficient for linear fits) to scientific questions and engineering problems, using digital tools when feasible.

5. Use simple limit cases to test mathematical expressions, computer programs, algorithms, or simulations of a process or system to see if a model “makes sense” by comparing the outcomes with what is known about the real world.

Apply concepts of statistics and probability (including determining function fits to data, slope, intercept, and correlation coefficient for linear fits) to scientific and engineering questions and problems.

Grade 8. Statistics and Probability 8.SP

Investigate patterns of association in bivariate data.

1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear as

Interpreting Categorical and

Quantitative Data

• Summarize, represent, and interpret data on a single count or measurement variable.

• Summarize, represent, and interpret data on two categorical and quantitative variables.

• Interpret linear models.

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying statistical experiments.

• Make inferences and justify conclusions from sample surveys, experiments and observational studies.

Conditional Probability and the Rules of Probability

• Understand independence and conditional probability and use them to interpret data.

• Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Using Probability to Make Decisions

• Calculate expected values and use them to solve problems.

• Use probability to evaluate outcomes of decisions.

STANDARDS FOR MATHEMATICAL PRACTICE

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for an express regularity in repeated reasoning.

Conditional Probability and the Rules of Probability S-CP

Understand independence and conditional probability and use them to interpret data.

Algebra I: Statistics and Probability

Interpreting Categorical and Quantitative

Data

• Summarize, represent, and interpret data on a single count or measurement variable.

• Summarize, represent, and interpret data on two categorical and quantitative variables.

• Interpret linear models.

Geometry: Statistics and Probability Conditional Probability and the Rules of Probability

• Understand independence and conditional probability and use them to interpret data.

• Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Using Probability to Make Decisions

• Use probability to evaluate outcomes of decisions

Statistics and Probability

Conditional Probability and the Rules of Probability S-CP

Understand independence and conditional probability and use them to interpret data.

Algebra II: Statistics and Probability

Interpreting Categorical and Quantitative Data

• Summarize, represent and interpret data on a single count or measurement variable.

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying statistical experiments.

• Make inferences and justify conclusions from sample surveys, experiments and observational studies.

Using Probability to Make Decisions

• Use probability to evaluate outcomes of decisions.