We introduce the basic concepts of probability and apply them to simple physical systems and everyday life. We will discover the universal nature of the central limit theorem and the Gaussian distribution for the sum of a large number of random variables and discuss its relation to why thermodynamics is possible. Because of the importance of probability in many contexts and the relatively little time it will take us to consider more advanced topics, our discussion goes beyond what we will need for the applications of statistical mechanics. Computer simulations will be used to explore these concepts. These simulations can be found by searching ComPADRE for Open Source Physics, STP, or Statistical and Thermal Physics.

Author: Harvey Gould
Posted: Feb 18, 2010 at 10:19PM

We agree and updated our explanation. Thank you.

Harvey Gould

> On Feb 02, 2010, jean-michel levy posted: > > Using Stirlin's > formula here is akin to smash a fly with a hammer > > It is enough to observe that the binomial coefficient > C_n^k can be rewritten: > > > n^k/k! [1(1-1/n)(1-2/n)..(1-(n-k)/n)] > The number of factors between brackets is fixed and > they all tend to 1 > when n--> infinity. > > Replacing > now p by l/n yields P[X=k] ~ n^k/k! (l/n)^k(1-l/n)^{n-k} > and the last factor --> exp(-l) in the > desired limit, > which completes the proof.

<a href="https://www.compadre.org/OSP/items/detail.cfm?ID=7272">Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." In Thermal and Statistical Physics. 2008.</a>

H. Gould and J. Tobochnik, in Thermal and Statistical Physics (2008), WWW Document, (https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446).

H. Gould and J. Tobochnik, STP Textbook Chapter 3: Concepts of Probability, in Thermal and Statistical Physics (2008), <https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446>.

Gould, H., & Tobochnik, J. (2008). STP Textbook Chapter 3: Concepts of Probability. In Thermal and Statistical Physics. Retrieved January 27, 2021, from https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446

Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." In Thermal and Statistical Physics. 2008. https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446 (accessed 27 January 2021).

Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." Thermal and Statistical Physics. 2008. 1 May 2008. 27 Jan. 2021 <https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446>.

@incollection{
Author = "Harvey Gould and Jan Tobochnik",
Title = {STP Textbook Chapter 3: Concepts of Probability},
BookTitle = {Thermal and Statistical Physics},
Month = {May},
Year = {2008}
}

%0 Book Section %A Gould, Harvey %A Tobochnik, Jan %D May 1, 2008 %T STP Textbook Chapter 3: Concepts of Probability %B Thermal and Statistical Physics %8 May 1, 2008 %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446

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