7 o6 THE POPULAR SCIENCE MONTHLY. 



Rule for the Multiplication of Probabilities. Given the separate 

 probabilities of the two consequences, " If A then B," and " If both 

 A and B, then C." Then the product of these two numbers is the 

 probability of the consequence, " If A, then both B and C." 



Special Mule for the Multiplication of Independent Probabilities. 

 Given the separate probabilities of two consequences having the same 

 antecedents, " If A, then B," and " If A, then C." Suppose that these 

 consequences are such that the probability of the second is equal to 

 the probability of the consequence, " If both A and B, then C." Then 

 the product of the two given numbers is equal to the probability of 

 the consequence, " If A, then both B and C." 



To show the working of these rules we may examine the proba- 

 bilities in regard to throwing dice. What is the probability of throw- 

 ing a six with one die ? The antecedent here is the event of throwing 

 a die ; the consequent, its turning up a six. As the die has six sides, 

 all of which are turned up with equal frequency, the probability of 

 turning up any one is ^. Suppose two dice are thrown, what is the 

 probability of throwing sixes ? The probability of either coming up 

 six is obviously the same when both are thrown as when one is thrown 

 namely, %. The probability that either will come up six when the 

 other does is also the same as that of its coming up six whether the 

 other does or not. The probabilities are, therefore, independent; and, 

 by our rule, the probability that both events will happen together is 

 the product of their several probabilities, or -|- x . What is the 

 probability of throwing deuce-ace ? The probability that . the first 

 die will turn up ace and the second deuce is the same as the proba- 

 bility that both will turn up sixes namely, -^ ; the probability that 

 the second will turn up ace and the frst deuce is likewise -fa ; these 

 two events first, ace ; second, deuce ; and, second, ace ; first, deuce 

 are incompatible. Hence the rule for addition holds, and the prob- 

 ability that either will come up ace and the other deuce is -^g- -f- -fa, 

 or^. 



In this way all problems about dice, etc., may be solved. When 

 the number of dice thrown is supposed very large, mathematics (which 

 may be defined as the art of making groups to facilitate numeration) 

 comes to our aid with certain devices to reduce the difficulties. 



II. 



The conception of probability as a matter of fact, i. e., as the pro- 

 portion of times in which an occurrence of one kind is accompanied 

 by an occurrence of another kind, is termed by Mr. Venn the mate- 

 rialistic view of the subject. But probability has often been regarded 

 as being simply the degree of belief which ought to attach to a 

 proposition ; and this mode of explaining the idea is termed by Venn 

 the conceptualistic view. Most writers have mixed the two conceptions 

 together. They, first, define the probability of an event as the reason 



