100

9

Probability and Likelihood

proposition that a necessary condition for an event to be causally conditioned is that

it can be predicted with certainty. If, however, we compare a prediction of a physical

phenomenon with more and more accurate measurements of that phenomenon, one is

forced to reach a remarkable conclusion—that in not a single instance is it possible

to predict a physical event exactly, unlike a purely mathematical calculation. The

“indeterminists” interpret this state of affairs by abandoning strict causality and

asserting that every physical law is of a statistical nature. The opposing school of

“determinists” asserts that the laws of nature apply to an idealized world-picture,

in which phenomena are represented by precise mathematical symbols, which can

be operated on according to strict and generally agreed rules and to which precise

numbers can be assigned (to which an actual measurement can only approximate);

in the corresponding mentally constructed world-picture, all events follow definable

laws and are strictly determined causally; the uncertainty in the prediction of an event

in the world of sense is due to the uncertainty in the translation of the event from the

world of sense to the world-picture and vice versa. It is left to the interested reader

to pursue the implications with respect to quantum mechanics (with which we shall

not be explicitly concerned in this book).

Sommerhoff (1950) formulated probability in the following terms: Given a system

whose initial state can be one of a setupper QQ ofnn alternativesupper Q 1 comma upper Q 2 comma ellipsis comma upper Q Subscript n BaselineQ1, Q2, . . . , Qn, of which

a certain fraction m divided by nm/n will lead to the subsequent occurrence of an event upper EE that

is to be expected in the normal development of the system, then the probability

that any particular member of upper QQ leads to upper EE is given by the fraction m divided by nm/n. Note

that this formulation only applies to the effects of the initial states, not to the states

themselves. It has the advantage of avoiding any assumption of equally probable, or

equally uncertain, events.

Before any further discussion about probability can take place, it is essential to

agree on what is meant by the possible results from an experiment (or observa-

tion). These results are called “events”. Very often abstract models, corresponding

to idealized events, are constructed to assist in the analysis of a phenomenon.

9.2

Fundamentals

The elementary unit in probability theory is the event. One has a fair freedom to define

the event; simple events are irreducible and compound events are combinations of

simple events. For example, the throw of a die to produce a 5 (with probability 1/6)

is a simple event, and combinations of events to yield the same final result, such as

three 2s, or a 5 and a 1, are compound events. Implicitly, the level of description

is fixed when speaking of events in this way; clearly, the “event” of throwing a 6

requires many “sub-events” (which are events in their own right) involving muscular

movements and nervous impulses, but these take place on a different level.