By Fazlollah M. Reza
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For a more thorough discussion the reader is referred to Cramer (Chap. 4, The Lebesgue Measure of a Linear Point Set). 8. of Events. * In Sec. 2-7 on Probability Measure an introductory axiomatic account of probability aa a measure of a set was this section is to supplement the set-theory given. The object of view with some perhaps less formal discussion of the probability of occurrence of certain events of a defined experiment. In other words, we wish to make a transition from the suggested abstract mathematical measure to some numerical function the measure requirements.
The coin is thrown twice. If a head appears in at least one of the two throws, A wins. Otherwise, B wins. Intuitively, it seems that the four following possible outcomes are equally probable. (HH) , (HT), (TH), (TT) where H denotes head and T denotes tail. A may assume that his chances of winning the game are %, since a head occurs in three out of four cases (to his advantage). On the other hand, the following reasoning may also seem logical. If the outcome of the first throw is H, A wins; there is no need to continue the game.
I~ILI-t1, tive diagram called a Venn * diagram is of considerable visual assistance. FIG. 2-1. Example of a Venn diagram. The elements of the universal set in a Venn diagram are generally shown by points in a rectangle. The elements of any set under consideration are commonly shown by a circle or by any other simple closed contour inside the universal set. The universe associated with the aforesaid is illustrated in 2-1. A set may contain a finite or an infinite numbe-r of elements. a set has no element, it is said to be an empty or a null set.