694 APPENDIX 



thousand people thirty thousand are vaccinated, and that five hundred per- 

 sons have smallpox. If vaccination has no influence on the number of cases 

 of smallpox, what is the probability that a person will be both vaccinated 

 and have smallpox ? 



Since one hundred thousand is a large number, we may give 



-^ = = probability that a person is vaccinated ; 



I 00000 10 



= probability that a person has smallpox ; 



i ooooo 200 

 3 i 3 



= probability that a person is both vaccinated 



10 200 2000 



and has smallpox. 



Furthermore, - x i ooooo = 150, the number of persons we should ex- 

 pect both to be vaccinated and to have smallpox, if vaccination has no 

 influence on the number of cases of smallpox. 



Illustrations of probability. Let us throw out upon a table at random four 

 pennies ; what is the probability that exactly r of them will be heads and the 

 rest tails when r takes values o, i, 2, j, 4 ? 



(1) Probability that o will be head and 4 tails is Q) 4 



(2) Probability that i will be head and 3 tails is 4(J) 4 



(3) Probability that 2 will be heads and 2 tails is 6(|) 4 



(4) Probability that 3 will be heads and i tail is 4(^) 4 



(5) Probability that 4 will be heads and o tail is (|) 4 



In (2) the coefficient 4 appears before Q) 4 because with four coins there 

 are four different combinations 1 possible, each consisting of i head and 3 

 tails. Similarly in (3) the coefficient 6 appears because with four coins 

 there are possible six combinations, each consisting of 2 heads and 2 tails. 



The above illustration may be generalized and the result put into the 

 following form : 



If n coins are thrown upon a table at random, the probability that exactly 

 r of them will be heads and the rest tails is given by the r+ ist term of 



the binomial expansion I- + - ) ; that is, in other symbols, "CV(?)" where 

 the symbol "C r indicates the number of combinations of n things taken r 

 at a time. 



In order that the reader may more fully appreciate the greater prob- 

 ability of getting an almost equal number of heads and tails in tossing 

 pennies than of getting widely different numbers, we present the following 

 table for n = 999> obtained from Que'telet Sur la thdorie des probability. 



1 For definition of a combination, see text, p. 511. 



