Likelihood  (Page 2/4)

Early developments of probability as a mathematical discipline, freeing it from its religious and magical overtones, came as a response to questionsabout games of chance played repeatedly. The mathematical formulation owes much to the work of Pierre de Fermat and Blaise Pascal in theseventeenth century. The game is described in terms of a well defined trial (a play); the result of any trial is one of a specific set of distinguishableoutcomes. Although the result of any play is not predictable, certain “statistical regularities” of results are observed. The possibleresults are described in ways that make each result seem equally likely. If there are N such possible “equally likely” results, each is assigned a probability $1/N$ .

The developers of mathematical probability also took cues from early work on the analysis of statistical data. The pioneering work of John Grauntin the seventeenth century was directed to the study of “vital statistics,” such as records of births, deaths, and various diseases. Graunt determined thefractions of people in London who died from various diseases during a period in the early seventeenth century. Some thirty years later, in 1693, Edmond Halley (forwhom the comet is named) published the first life insurance tables. To apply these results, one considers the selection of a member of the population on achance basis. One then assigns the probability that such a person will have a given disease. The trial here is the selection of a person, but theinterest is in certain characteristics. We may speak of the event that the person selected will die of a certain disease– say “consumption.”Although it is a person who is selected, it is death from consumption which is of interest. Out of this statistical formulation came an interest not only inprobabilities as fractions or relative frequencies but also in averages or expectatons. These averages play an essential role in modern probability.

We do not attempt to trace this history, which was long and halting, though marked by flashes of brilliance. Certain concepts and patterns which emergedfrom experience and intuition called for clarification. We move rather directly to the mathematical formulation (the “mathematical model”) whichhas most successfully captured these essential ideas. This is the model, rooted in the mathematical system known as measure theory, is called the Kolmogorov model , after the brilliant Russian mathematician A.N. Kolmogorov (1903-1987). Kolmogorov succeeded in bringing together variousdevelopments begun at the turn of the century, principally in the work of E. Borel and H. Lebesgue on measure theory. Kolmogorov published his epochal work in German in 1933. It was translatedinto English and published in 1956 by Chelsea Publishing Company.

Outcomes and events

Probability applies to situations in which there is a well defined trial whose possible outcomes are found among those in a given basic set. The following are typical.

• A pair of dice is rolled; the outcome is viewed in terms of the numbers of spots appearing on the top faces of the twodice. If the outcome is viewed as an ordered pair, there are thirty six equally likely outcomes. If the outcome is characterized by the total number of spots on the twodie, then there are eleven possible outcomes (not equally likely).
• A poll of a voting population is taken. Outcomes are characterized by responses to a question. For example, the responses may be categorized as positive (or favorable), negative (or unfavorable), oruncertain (or no opinion).
• A measurement is made. The outcome is described by a number representing the magnitude of the quantity in appropriate units. In some cases, the possible values fall amonga finite set of integers. In other cases, the possible values may be any real number (usually in some specified interval).
• Much more sophisticated notions of outcomes are encountered in modern theory. For example, in communication or control theory, a communication system experiences only one signalstream in its life. But a communication system is not designed for a single signal stream. It is designed for one of an infinite set of possible signals. The likelihood of encountering a certain kind of signal is important in the design. Suchsignals constitute a subset of the larger set of all possible signals.