MEASUREMENT OF VARIABLE PROPERTIES 123 



The order of succession being neglected, the sixty-four events 

 are reduced to seven, which are represented by the terms 

 obtained by expanding (^ + 6)^ in which a = b = i — viz. 



(a + b)^ = a^ + 6a^b + i^a^b^ + 2oa%^ + iSa^ft* + 6ab^ + b^ 



About the information given by these terms, see p. 120, 

 {i)-(7), and compare with the above table. 



QUESTION I. : Which is the frequency of the event a*b^ (four heads and 

 two tails) ? 



Answer : 15 : 64 ( = 0-234). 



QUESTION II. : In a series of x tosses the event ab^ has been observed 

 300 times. How many times did the event a^b^ occur ? 



Answer : Approximately 750 times. 



QUESTION III. : Which is in Question II. the value of x ? 



Answer : Approximately 3200 tosses. 



QUESTION IV. : The number of tosses being 200, how many times will the 

 event a^ be observed ? 



Answer : Approximately three times (this answer has hardly any practical 

 value, because the number of tosses is too small) . 



QUESTION V. : The number of tosses being 6000, in how many tosses will 

 the number of events a (heads) be an even number ? 



Answer : Approximately 2906 times. 



§97.— REMARKS ABOUT THE PRECEDING EX- 

 AMPLES. — Chance is a vague something which is continually 

 varying. At first sight the effects of chance seem to be 

 capricious, as if they were independent of any rule whatever. 



In the example of the spherical ball (§91), however, all the 

 possible effects of chance are reduced to one simple event, the 

 occurrence of which is certain. 



Tossing one coin (§92), we have reduced all the possible effects 

 of chance to two simple events a and b. The frequency of each 

 of them is |. The sum of their frequencies (probabilities) is i. 

 The figure i is the expression of certitude. If we consider one 

 toss, we may foretell with certitude that aor b will be observed, 

 but we do not know at all which of both will happen : the notion 

 of frequency (probability) is in this case a pure fiction. But 

 in proportion as the tosses become more numerous the exist- 

 ence of a rule becomes more apparent. We learn from EX- 

 PERIENCE that the frequencies of a and b approach more and 

 more to equality, which is represented by their frequencies 

 (I and J) calculated a priori. 



The events a and b afford us a fixed starting-point for further 

 investigation. We look upon them as being simple causes (or 

 forces) the combinations of which produce combined causes 

 (resultants). We have succeeded in discovering certain rules 

 by which the combined causes and their effects are governed — 

 whatever may be the number n of simple causes (simple events) 

 which arc combined. 



The rules discovered are rules (or laws) of chance. The 



