Probability

IB Higher Level Mathematics : Core

7. Probability

7.1

The basic notions of probabilities of events in a sample space, finite and infinite.

See Sample Spaces and Events, Introduction to Probability, and Principles of Probability, on-line tutorials accompanying the book Finite Mathematics & Calculus Applied to the Real World, by Stefan Waner and Steven Costenoble of Hofstra University.

The Statistics Glossary contains many definitions used in the IB Higher Level Mathematics Probability unit.

For a free, complete and downloadable textbook, see Introduction to Probability, by Charles M. Grinstead, of Swarthmore, and J. Laurie Snell, of Dartmouth. The book may be downloaded in Adobe Portable Document Format (PDF), or in Postscript Format (ps). The PDF version requires Adobe Acrobat Reader 3.0, also free and downloadable. The entire book may be dowloaded at once, or by individual chapter. Topics covered include conditional probability, discrete and continuous probability, expected value, and variance. Answers to odd-numbered questions are also given.

7.2

Tree diagrams and Venn diagrams.

See Venn Diagrams and Regions in a Venn Diagram, both from California State University San Bernardino's Reference Notes Page.

7.3

Mutually exclusive events. The probability of the union of two events that are not mutually exclusive.

For three different explanations of mutually exclusive events, see (1) Mutually Exclusive Events, from the Probability section of the Statistics Glossary; (2) Mutually exclusive events, from the Probability chapter of HyperStat Online; and (3) Mutually Exclusive, from Eric's Treasure Trove of Mathematics.

7.4

Independence.

For an explanation of the concept, see Independent Events, from the Probability section of the Statistics Glossary.

7.5

Conditional probability.

See Conditional probability, from HyperStat Online.

7.6

Derivation and use of Bayes' Theorem for two events.

A downloadable program dealing with probability, conditional probability and Bayes' Theorem is available from Bayes' Theorem. The program is a part of the Mathematical Toolkits Software developed at the University of Arizona.

7.7

Discrete probability distributions. Expectation, mode, median, variance and standard deviation.

See Introduction to Probability Models - Random Variables for an explanation of the concept, together with an example in the form of a JAVA applet. Also see Random Variables, part of the electronic textbook Statistics : The Study of Stability in Variation, from UCLA.

For background material, see Variance and Standard Deviation, one of several Introductory Statistics Demonstrations from Ohio University's Department of Psychology. The site includes a demonstration of the variance formula.

7.8

The binomial distribution. Its mean and variance, without proof. Applications.

For three pages of background information, see Binomial distribution, from Rice University's HyperStat Online site.

A downloadable demo of the binomial probability distribution is one of several programs available from the University of Western Ontario. Click Probability and Statistics Programs.

7.9

Applications of the hypergeometric distribution.

7.10

Continuous probability distributions. Expectation, mode, median, variance and standard deviation.

See Continuous Random Variables and Histograms, and Mean, Median, Variance and Standard Deviation. Both links are on-line resources for the text Calculus Applied to Probability and Statistics for Liberal Arts and Business Majors.

The median, M, of a continuous probability distribution. See Mean, Median, Variance and Standard Deviation, one of the sections from Calculus Applied to Probability and Statistics, from Hofstra University. Used with permission.

7.11

The normal distribution. Standardization of a normal distribution and use of the standard normal distribution table.

See The Normal Density, part of the Statistics : The Study of Stability in Variation site from UCLA

Also see Probability Density Functions : Uniform, Exponential, Normal, and Beta, Section 2 of Calculus Applied to Probability and Statistics for Liberal Arts and Business Majors, a "complete text resource" written by Stefan Waner and Steven Costenoble. This on-line text is part of Hofstra University's extensive Finite Mathematics & Applied Calculus site.

Background material is provided at Z-Distributions and Z-Scores and Probability, two of several Introductory Statistics Demonstrations from David Wallace of the Department of Psychology at Ohio University.

Chapter 5 : Normal Distribution, from HyperStat Online, also provides background material from the point of view of a psychologist.

Using the Normal Distribution Calculator, z-scores are entered, and the standard normal probability that a score lies between these values is calculated and graphed. A similar demo can be found at Probability and Quantile Applets, with the refinement that probabilities can be calculated to the left and to the right of the entered values, as well as between them. Both applets require a JAVA-enabled browser.

7.12

The normal approximation to the binomial distribution. Continuity corrections.

For an interactive demonstration, see the normal approximation to the binomial distribution applet from Rice University's HyperStat Online site (below). Recommended.

With the probability of success on any one of the 20 trials being 0.3, the binomial
distribution is fairly closely approximated by the normal distribution. The height of each segment represents the probability of getting the number of successes indicated on the horizontal axis. The mode is 6.

The probability of success on any one of the 20 trials has been lowered to 0.2. The normal curve does not provide as close an approximation to the binomial distribution as in the case above. The mode is now 4.

This interactive applet, from the HyperStat Online site, allows the user to change the number of trials, as well as the probability of success on each trial. Used with permission.