My favorite of such applications of probability is the Poisson stochastic arrival (get arrivals at discrete points in time) process. There are two amazing, powerful, useful, non-obvious biggie points:
(1)Axiomatic Derivation
As in
Erhan Çinlar, 'Introduction to Stochastic Processes', ISBN 0-13-498089-1.
an arrival process with stationary (distribution not changing over time) independent (of all past history of the process) increments (arrivals) is necessarily a Poisson process. So there is an arrival rate, and times between arrivals are independent, identically distributed exponential random variables.
Often in practice can check these assumptions well enough just intuitively.
Then as in Çinlar can quickly derive lots of nice, useful results.
(2)The Renewal Theorem.
As in
William Feller,
'An Introduction to Probability Theory and
Its Applications,
Second Edition,
Volume II',
ISBN 0-471-25709-5,
roughly, with meager assumptions and approximately, if the arrivals are from many different independent sources, not necessarily Poisson, then the resulting process, that is, the sum from the many processes, is Poisson.
E.g., using (2), between 1 and 2 PM, the arrivals at a busy Web site, coming from lots of independent Web users, will look Poisson and from (1) can say a lot for sizing the server farm, looking for DDOS attacks, security, performance, network, and system management anomalies, etc., e.g., do statistical hypotheses tests.
Similarly for packets on a busy communications network, server failures in a server farm, etc.
(1)Axiomatic Derivation
As in
Erhan Çinlar, 'Introduction to Stochastic Processes', ISBN 0-13-498089-1.
an arrival process with stationary (distribution not changing over time) independent (of all past history of the process) increments (arrivals) is necessarily a Poisson process. So there is an arrival rate, and times between arrivals are independent, identically distributed exponential random variables.
Often in practice can check these assumptions well enough just intuitively.
Then as in Çinlar can quickly derive lots of nice, useful results.
(2)The Renewal Theorem.
As in
William Feller, 'An Introduction to Probability Theory and Its Applications, Second Edition, Volume II', ISBN 0-471-25709-5,
roughly, with meager assumptions and approximately, if the arrivals are from many different independent sources, not necessarily Poisson, then the resulting process, that is, the sum from the many processes, is Poisson.
E.g., using (2), between 1 and 2 PM, the arrivals at a busy Web site, coming from lots of independent Web users, will look Poisson and from (1) can say a lot for sizing the server farm, looking for DDOS attacks, security, performance, network, and system management anomalies, etc., e.g., do statistical hypotheses tests.
Similarly for packets on a busy communications network, server failures in a server farm, etc.