Probability, Random Variables, Statistics, and Random Processes

2Applications in Probability

Noting there are numerous applications in probability, our focus here is on just a few applications. In this chapter, some major probability applications in science and engineering, such as systems reliability, medical diagnostic testing, and Bayesian spam filtering, as well as some interesting probability applications in life, such as odds and risk, gambler's ruin problem, and how to make the best choice problem, are briefly highlighted.

2.1 Odds and Risk

In the world of gambling, games of chance, and sports, probabilities are often expressed by odds. Odds are also involved in the statistical analysis of frequency data, such as in medicine and business. Odds measures are calculated when the study is either retrospective (backward looking), such as the impacts of various treatments in medicine, or prospective (forward looking), such as the prospects of teams winning in sports events.

The odds for (in favor of) an event A, which are the ratio of the probability of A to the probability of A^c, reflect the likelihood that the event A will take place. The odds against (not in favor of) an event A, which are the ratio of the probability of A^c to the probability of A, highlight the likelihood that the event A will not take place. We thus have the following:

2.1

For a given odds, the probability of the event A can be thus easily determined. Note that the ...