On the Continuity of Chance 9 



ten ways: for any one of the ten coins may be the one showing 

 tail. So nine heads, one tail is ten times as probable as all heads. 

 The probabilities for the eleven different happenings are 

 proportional to the number of ways respectively in which 

 nothing, one thing, two things, . . . ., ten things can be picked 

 out from ten things. These are also the coefficients in the 

 expansion of a binomial to the tenth power; so that we can 

 say that the probability that a heads, h tails (o + 6 =10) in 

 tossing ten coins is proportional to the coefficient of A " i in 

 the expansion of (/i+^)'"- The sum of these coefficients is 

 (1 + 1) '"=1024. So in 1024 throws, in the absence of further 

 knowledge, we must regard as probable about as follows: 



So the chances are only 252 in 1024, or less than 1 in 4, 

 that we shall get half heads, half tails, although this is the 

 most probable throw. The probability, however, that a throw 

 does not vary from this more than by the turning of one coin 

 is 072 in 1024, or nearly 7 in 10. 



Acrain, notice the about. We can never be certain in mat- 

 ters that chance controls; or, if you prefer, in our confessed 

 ignorance to assert certainty would be gross presumption. 



That such numbers of throws are probable means merely 

 that if we threw set after set, millions on millions of sets, of 

 1024 throws, then the average numbers for each sort of result 

 would closely approximate what we have given, the excesses 

 in some sets being cancelled by defects in others. In the 

 same way, in the above calculation, the numbers of throws in 

 which there are more heads than tails balance those in which 

 there are more tails than heads. 



By an extension of the method used to get probable num- 

 bers of different sorts of throws we could get the numbers of 



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