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Certain concepts and patterns have emerged from experience and intuition. The mathematical formulation (the mathematical model) which has most successfully captured these essential ideas is rootedin measure theory, and is known as the Kolmogorov model , after the brilliant Russian mathematician A.N. Kolmogorov (1903-1987).

One cannot prove that a model is correct . Only experience may show whether it is useful (and not incorrect). The usefulness of the Kolmogorov model is establishedby examining its structure and showing that patterns of uncertainty and likelihood in any practical situation can be represented adequately. Developments, such as in this course, have given ample evidence of such usefulness.

The most fruitful approach is characterized by an interplay of

  • A formulation of the problem in precise terms of the model and careful mathematical analysis of the problem so formulated.
  • A grasp of the problem based on experience and insight. This underlies both problem formulation and interpretation of analytical results of the model. Often such insight suggestsapproaches to the analytical solution process.

Matlab: a tool for learning

In this work, we make extensive use of MATLAB as an aid to analysis. I have tried to write the MATLAB programs in such a way that they constitute useful,ready-made tools for problem solving. Once the user understands the problems they are designed to solve, the solution strategies used, and the manner in which thesestrategies are implemented, the collection of programs should provide a useful resource.

However, my primary aim in exposition and illustration is to aid the learning process and to deepen insight into the structure of the problems considered and the strategies employed in their solution. Several features contribute to that end.

  1. Application of machine methods of solution requires precise formulation. The data available and the fundamental assumptions must be organized in an appropriate fashion.The requisite discipline for such formulation often contributes to enhanced understanding of the problem.
  2. The development of a MATLAB program for solution requires careful attention to possible solution strategies. One cannot instruct the machine without a clear grasp of what isto be done.
  3. I give attention to the tasks performed by a program, with a general description of how MATLAB carries out the tasks. The reader is not required to trace out all theprogramming details. However, it is often the case that available MATLAB resources suggest alternative solution strategies. Hence, for those so inclined, attention to the details maybe fruitful. I have included, as a separate collection, the m-files written for this work. These may be used as patterns for extensions as well as programs in MATLAB for computations. Appendix A provides a directory of these m-files.
  4. Some of the details in the MATLAB script are presentation details. These are refinements which are not essential to the solution of the problem. But they makethe programs more readily usable. And they provide illustrations of MATLAB techniques for those who may wish to write their own programs. I hope many will be inclined togo beyond this work, modifying current programs or writing new ones.

An invitation to experiment and explore

Because the programs provide considerable freedom from the burden of computation and the tyranny of tables (with their limited ranges and parameter values), standardproblems may be approached with a new spirit of experiment and discovery. When a program is selected (or written), it embodies one method of solution.There may be others which are readily implemented. The reader is invited, even urged, to explore! The user may experiment to whatever degree he or shefinds useful and interesting. The possibilities are endless.


After many years of teaching probability, I have long since lost track of all those authors and books which have contributed to the treatmentof probability in this work. I am aware of those contributions and am most eager to acknowledge my indebtedness, althoughnecessarily without specific attribution.

The power and utility of MATLAB must be attributed to to the long-time commitment of Cleve Moler, who made the package available in the public domain for several years. The appearanceof the professional versions, with extended power and improved documentation, led to further appreciation and utilization of its potential in applied probability.

The Mathworks continues to develop MATLAB and many powerful "tool boxes," and to provide leadership in many phases of modern computation. They have generously made available MATLAB 7to aid in checking for compatibility the programs written with earlier versions. I have not utilized the full potential of this version for developing professional quality user interfaces, since Ibelieve the simpler implementations used herein bring the student closer to the formulation and solution of the problems studied.


The development and organization of the CONNEXIONS modules has been achieved principally by two people: C.S.(Sid) Burrus a former student and later a faculty colleague, then Dean of Engineering, and most importantly a long time friend; and Daniel Williamson, a music major whose keyboard skills have enabled him to set up the text (especially the mathematical expressions) with great accuracy, and whose dedication to the task has led to improvements in presentation. I thank them and others of the CONNEXIONS team who have contributed to the publication of this work.

Paul E. Pfeiffer

Rice University

Questions & Answers

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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