Overview¶
Probability will be the way that we quantify how likely something is to occur (in the sense of one of the interpretations in 1.2 Interpretations of Probability). In this section, we give examples of the types of situations in which probability will be used.
1.3.1 Types of Experiments¶
The theory of probability pertains to the various possible outcomes that might be obtained and the possible events that might occur when an experiment is performed.
The breadth of this definition allows us to call almost any imaginable process an experiment whether or not its outcome will ever be known. The probability of each event will be our way of saying how likely it is that the outcome of the experiment is in the event. Not every set of possible outcomes will be called an event. We shall be more specific about which subsets count as events in 1.4 Set Theory.
Probability will be most useful when applied to a real experiment in which the outcome is not known in advance, but there are many hypothetical experiments that provide useful tools for modeling real experiments. A common type of hypothetical experiment is repeating a well-defined task infinitely often under similar conditions.
Some examples of experiments and specific events are given next. In each example, the words following “the probability that” describe the event of interest.
In an experiment in which a coin is to be tossed 10 times, the experimenter might want to determine the probability that at least four heads will be obtained.
In an experiment in which a sample of 1000 transistors is to be selected from a large shipment of similar items and each selected item is to be inspected, a person might want to determine the probability that not more than one of the selected transistors will be defective.
In an experiment in which the air temperature at a certain location is to be observed every day at noon for 90 successive days, a person might want to determine the probability that the average temperature during this period will be less than some specified value.
From information relating to the life of Thomas Jefferson, a person might want to determine the probability that Jefferson was born in the year 1741.
In evaluating an industrial research and development project at a certain time, a person might want to determine the probability that the project will result in the successful development of a new product within a specified number of months.
1.3.2 The Mathematical Theory of Probability¶
As was explained in 1.2 Interpretations of Probability, there is controversy in regard to the proper meaning and interpretation of some of the probabilities that are assigned to the outcomes of many experiments. However, once probabilities have been assigned to some simple outcomes in an experiment, there is complete agreement among all authorities that the mathematical theory of probability provides the appropriate methodology for the further study of these probabilities. Almost all work in the mathematical theory of probability, from the most elementary textbooks to the most advanced research, has been related to the following two problems: (i) methods for determining the probabilities of certain events from the specified probabilities of each possible outcome of an experiment and (ii) methods for revising the probabilities of events when additional relevant information is obtained.
These methods are based on standard mathematical techniques. The purpose of the first six chapters of this book is to present these techniques, which, together, form the mathematical theory of probability.