I'm reading a graduate book on measure theoretic probability theory by Cinlar. In the introduction he says

The actual assignment of probabilities to events is the primary task of the probabilist. It requires much thought and experience, it is rarely explicit, and it determines the quality of the probability space as a model of the experiment involved. Once the probability space is fixed, the main task is to evaluate various integrals of interest by making adroit use of those implicitly defined probabilities. Often, the results are compared against experience, and the probability space is altered for a better fit.

I'm wondering: Is this actually how it's done in practice? Can you give me concrete example of real-life problems (expressed as the mentioned "integrals") that are solved this way?

I'm a pure mathematician, rooted in theory, and since Cinlar doesn't give any further explanation, I'm skeptical of his statement.

I was always living under the impression that the whole framework of probability theory was NOT developed in order to allow people to solve real-life problems in the way Cinlar described it - but rather for special use cases, like stochastic processes or statistics.


Probability is an area of study which involves predicting the relative likelihood of various outcomes. It is a mathematical area that has developed over the past three or four centuries. One of the early uses was to calculate the odds of various gambling games. Its usefulness for describing errors of scientific and engineering measurements was soon realized. Engineers study probability for its many practical uses, ranging from quality control and quality assurance to communication theory in electrical engineering. Engineering measurements are often analyzed using statistics, Â and a good knowledge of probability is needed in order to understand statistics.

Statistics is a word with a variety of meanings. To the man in the street, it most often means simply a collection of numbers, such as the number of people living in a country or city, a stock exchange index, or the rate of inflation. These all come under the heading of descriptive statistics, in which items are counted or measured and the results are combined in various ways to give useful results. That type of statistics certainly has its uses in engineering. But another type of statistics will engage our attention to a much greater extent. That is inferential statistics or statistical inference. For example, it is often not practical to measure all the items produced by a process. Instead, we very frequently take a sample and measure the relevant quantity on each member of the sample. We infer something about all the items of interest from our knowledge of the sample. A particular characteristic of all the items we are interested in constituting a population. Measurements of the diameter of all possible bolts as they come off a production process would make up a particular population. Â

Application of Probability and Statistics in Science:

Research Design The use of statistical methods and probability tests in research is an important aspect of science that adds strength and certainty to scientific conclusions. For example, in 1843, John Bennet Lawes, an English entrepreneur, founded the Rothamsted Experimental Station in Hertfordshire, England to investigate the impact of fertilizer application on crop yield. Lawes was motivated to do so because he had established one of the first artificial fertilizer factories a year earlier. For the next 80 years, researchers at the Station conducted experiments in which they applied fertilizers, planted different crops, kept track of the amount of rain that fell, and measured the size of the harvest at the end of each growing season.Â

By the turn of the century, the Station had a vast collection of data but few useful conclusions: One fertilizer would outperform another one year but underperform the next, certain fertilizers appeared to affect only certain crops, and the differing amounts of rainfall that fell each year continually confounded the experiments (Salsburg, 2001). The data were essentially useless because there were a large number of uncontrolled variables.Â

Computer Science Probability and Statistics for Computer Science develops introductory topics in probability with this particular emphasis, providing computer science students with an invaluable resource in their continued studies and professional research.Â

Probability and Statistics for Computer Science treat the most common discrete and continuous distributions, showing how they find use in decision and estimation problems, and constructs computer algorithms for generating observations from the various distributions. This one-of-a-kind resource also: Includes a thorough and rigorous development of all the necessary supporting mathematics Provides an opportunity to reconnect applications with the theoretical concepts of distributions introduced in prerequisite courses Gathers supporting topics in an appendix: set theory, limit processes, real number structure, Riemann-Stieltjes integrals, matrix transformation, and determinants Uses computer science examples such as client-server performance evaluation and image processing The author also addresses a variety of supporting topics, such as estimation arguments with limits, properties of power series, and Markov processes. Johnson's text proves an ideal resource for computer science students and practitioners interested in a probability study specific to their field.Â

Environmental Engineering

Issues related to environmental risk assessment include health effects, impact on natural resources or man-made structures due to pollution, change in climatic conditions, water quality of streams, etc. There are different parametric, non-parametric and empirical models are used to address these issues. Probability methods play a role in: Estimation of model parameters. Identification of probability distribution. Determination of dependencies among variables. Estimation of model uncertainties etc.

Geotechnical Engineering

In geotechnical engineering, there are different sources of uncertainty. For instance, the variable nature of the characteristics of rock affects the load bearing capacity. Heterogeneous soil properties and other in-situ conditions are also uncertain. Due to the inherent heterogeneity of the characteristics of soil and rock, probabilistic methods are essential to compute the bearing. Uncertainties are assessed through basic probabilistic analyses and statistics, such as histogram analysis, the sample mean, variance, standard deviation, Coefficient of Variance (CV) and Probability Density Function (PDF), etc. These methods are very useful for estimation of in-situ properties from limited soil samples and for comparison of the field test to field performance data. Reliability of design and construction methods is also assessed in a probabilistic way. Moreover, the use of probability methods is inevitable to carry out the tradeoff analysis between the cost and benefits of proposed design strategies adopted in geotechnical engineering.


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