44 ESSAY ON PROBABILITIES. 



that the happening of one has nothing to do with that 

 of the other, the probability that both will happen is 

 the product of the probabilities that each will happen. 



This rule applies to any number of independent 

 events. 



It is indifferent whether the events are to happen to- 

 gether, or one after the other : thus the chances of a 

 compound drawing out of two lotteries, one drawing 

 out of each, are the same whether two persons draw 

 simultaneously, or one person first draws out of one, and 

 then out of the other. 



EXAMPLE. Let there be three lotteries, as follows : 

 f 6 white! f 7 white "I f 8 white! 

 \ 5 black J \ 2 black J \ 1 black J 



What is the chance of drawing from the three, white, 

 black, and white ? The probabilities of these events are, 

 _6_^ ^ an( j _&_ } the product of which is ^V^, or about 1 7 

 to 1 against the compound event. 



The probability that A will happen, and B will not, 

 is the product of the chance for A, and that against B. 

 The probability that one will happen, and one only, is 

 the sum of the probabilities 1. that A will happen and 

 B will not ; 2. That A will not happen and B will 

 happen. The probability that one or both will hap- 

 pen is the remainder when the probability that neither 

 will happen is subtracted from unity. The following 

 are evident results of the measure of probability : 



1. When either P, Q, or R must happen, the sum of 

 their probabilities must be unity, which is always the 

 representative of certainty. Thus, if a lottery contain 7 

 white, 5 black, and 3 red, either white, black, or red 

 must be drawn, and -fa, -j 5 3 , and -fa, together make 1, 

 and the same if the events be more or less than three in 

 number. 



2. When of two events, each excludes the other, 

 the probability that one or other of them will happen is 

 the sum of their probabilities. For to say that each 

 excludes the other, is to say that they are connected 

 events, or different possible cases of the same set. Thus, 



