1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

for stating that Ω consists of elements ω 1 , ω 2 , ..., ω n , ... Elements ω i arealso called elementary events or outcomes.Axiom. The sum of the probabilities of all the elementary events is1:P(ω 1 ) + ... + P(ω n ) + ... =∞∑ ∑P(ω ) = P(ω ) = 1.ii= 1 ω ⊆ ΩiiDefinition. An event is any subset (part of set) of the set ofelementary events; the probability of an event is the sum ofprobabilities of its elementary events. That set A is a subset of set Ω (i.e., that A consists of some elements included in Ω) is written asA ⊆ Ω . The probability of event A is denoted by P(A) and thedefinition is written down asP(A) =∑ω ⊆ AiP(ω ).iThe explanation below the symbol of summing means that those andonly those P(ω i ) are summed which are included in A.The described mathematical model can be applied for very manystochastic problems. However, all of them are initially formulated notin the terminology of the space of elementary events, i. e., not in theaxiomatic language but in ordinary terms. This [?] is unavoidablebecause only by considering problems any student of probabilitybecomes acquainted with those concrete situations in which it isapplicable. It is impossible to describe such situations in the axiomaticlanguage and it is therefore necessary to learn how to translate theconditions of problems into the language of elementary events.The situation here is quite similar to that which school studentsencounter when solving problems in compiling systems of equations:there, a translation from one language into another one is also needed.Such translations can be either very easy or difficult or ambiguouswith differing systems of equations appearing in the same problem. Inthis last-mentioned case, one such system can be difficult to compilebut easy to solve with the alternative system being opposite in thatsense (easy and difficult respectively).We stress therefore that, introducing a space of elementary eventscorresponding to a given problem, is not a purely mathematicaloperation as a proof of a theorem, but indeed a translation from onelanguage into another one, and it is senseless to strive for such a rigouras adopted in mathematics. Clear-cut mathematical formulations arenow concluded here and we are turning to the rules of translation.Stochastic problems usually have to do with some experiments, withthe set Ω consisting of all its possible outcomes. Thus, in coin tossingΩ consists of two elementary outcomesΩ = {heads; tails}and in throwing a die there are six such outcomes13