Prokop‘s chosen area of study involves a fair amount of physics: hisdaydreams have been filled with colliding gas molecules. While it maybe impossible to predict how single molecules will collide, it is possible topredict how great masses of colliding molecules will behave and hence thebehaviour of the gas as a whole. He has learned how the same mathematicaltechniques used in gas mechanics are also used in a multitude of explana-tions ranging from the stability of the orbits of planets to the spread of diseaseand the speed of losses at roulette tables. - P2
1.2 Von Mises‘s Relative FrequencyInterpretation
(중략) This is unfortunate, given howphilosophically loaded these terms are, and it would be best to avoid them.
That, however, is simply not possible, and so we must labour under them.
The standard theory of objective probability, as in the theory most likelyto be vaguely invoked in introductory texts to probability or statistics, isthe relative frequency interpretation. - P2
1.2.1 Probability and mass phenomena
Relative frequency theories are designed to deal with what von Mises calledmass phenomena. These are phenomena (or, better, types of outcomes ofobservations) that occur in very great quantities. This may be either becausewe can produce as many as we want by, for example, undertaking an experiment, or because there are a great many instances of them to be found. - P2
One of von Mises‘s examples was the probability of the declaration of war between Germany and Liberia. Liberia has declaredwar on Germany in both World Wars (reported in the New York Times on8 August 1917 and 28 January 1944. I don‘t know if Germany reciprocated). - P3
There is, however, no sharp boundary between mass and non-massphenomena. A borderline case, according to von Mises, is the question of the reliability of witnesses, presumably in jury trials: ‘We classified there liability and trust worthiness of witnesses and judges as a borderline case since we may feel reasonable doubt whether similar situations occur sufficiently frequently and uniformly for them to be considered as repetitivephenomena‘ (von Mises 1957: 10). - P3
Von Mises‘s interpretation of probability is meant to ground a scientifictheory of certain types of phenomena, namely, mass phenomena, and isto stand or fall with the confirmation of its applicability in practice. - P3
1.2.2 Convergence of relative frequency
Consider the humble coin toss. For any given number of flips of the coin,
a certain percentage will be heads. We can use the percentage of the occur-rence of heads in the total number of tosses as a measure of likeliness.
If heads are very likely, then the percentage of heads will be near 100 percent (nearly 100 out of 100, i.e. 1), that is, heads will turn up at nearly everytoss; if they are very unlikely, then the percentage will be nearer 0 (nearer 0 out of 100, i.e. none). If it is just as likely that the coin lands heads as tails(i.e. the coin is fair), then the ratio should be one-half. The name for theratio of heads to total tosses is the relative frequency of heads. - P4
Results of an experiment that we are looking for are called successes, otherwise failures. So, if we are flipping a coin and are looking forheads, we call the occurrence of a head a success. We could, of course, taketails to be success: nothing hinges on the name. These elementary events canbe combined into compound events, such as two tails occurring in a row, orthe tossing of two coins together, or the roll of a die and the toss of a coin. - P5
For coin tosses, the infinite series might look like HTHHTTH...
The sample space is then the set of all these infinite sequences. A collective(in German Kollektiv) is a member of the sample space (that is, an infinitesequence of outcomes) that obeys certain restrictions to be introduced shortly. - P6
The first axiom of von Mises‘s theory is the axiom of convergence: for a givencollective, the limiting relative frequency actually exists. This means thatthe value of the relative frequency does not oscillate, but settles down to some value. - P6
1.2.3 Randomness-the impossibility of a gambling system
The notion of limiting relative frequency was not original to von Mises.
What was original to his treatment was his notion of randomness. As noted earlier, another key feature of mass phenomena is that it is not possibleto predict individual outcomes-true mass phenomena are unpredictable.
The clearest examples can be found in the gambling hells. A quick searchwill reveal that there are many betting systems for various casino games andfor roulette in particular. A simple example of such a system is: bet on blackafter the ball has landed on red three times in a row. Following a systemlike this is a sure road to ruin, or at least to losing at the roulette table, as youcan confirm at your local casino. - P7
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