Author Richard Serfozo

Basic Probability Problems
May 20, 2003

Springer
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1 Elementary Concepts

The subject of applied probability appears to be rather fragmented because problems involving randomness arise in many different contexts and they often require the use of ad hoc mathematical techniques. The subject has more underlying structure, however, than meets the eye. The set of notes before you describes this structure via a number of basic probability problems. These notes are intended as a “supplement” to an introductory probability textbook that describes the methodology and notation. The basic problems presented here should be viewed as a follow-on to elementary, motivating examples. A reader just learning probability should aim at mastering these basic problems in the sense of being able to recognize them in various settings and solve them by carry out the required analysis and computations. A good way to study each problem is to create and solve one or more examples of the problem analogous to those below. By creating new examples this way, one will actually own a piece of the subject as well as understanding it.

1.1 Probabilities of Events
A description of events and their probabilities requires a framework for representing events of interest in terms of a random experiment. Our textbook tells us how to describe such an experiment in terms of a set of outcomes Ω called a sample space, and to represent events as subsets of Ω. A basic problem in this regard is as follows. Problem 1 Sample Spaces and Events as Sets Given a verbal description of a random experiment and certain events of interest, represent the sample space as a set Ω, and identify the events as subsets of Ω. There may be several natural choices for Ω; select the simplest one that contains enough information to describe