10 ON PROBABILITY. 
It may be remarked that if the same horse as Rapid Rhone were to run 
in both races, and if the probability of his winning the first race were 1, and 
of his winning the second 1; the probability of his winning both is not 1, 
but rather more. This very important distinction will be more fully ex- 
plained in treating of probabilities a posteriori. 
17. Let P,, @, be any two conflicting events, of which the probabilities 
are p,, ¢. 
The probability of the concurrence of all the events P,, P,,.. P,, is p, 
%X Ps.. P,, and the probability of any other event is that term in the product Ths 
P+q) Pq) Put q0); if 
in which the indices of p and q are the same as those of P and Q in the 
composite event which is considered. fos 
If we consider repeated trials of the same event, so that P,, P,, ... P, are i 
all the same, (p, +q) (Pa + @) -..-.... (Pu + ¢.) becomes (p 4 ¢), and poset 
the probability of any event composed of a times P and b times @ in any “i 
given order is p®¢’, the probability of having @ times P and b times Q, without ololy 
any regard to the order in which they occur, is the sum of the terms which are : ol 
equal to p® ¢’ in the developement of (p + ¢)", or, what is the same thing, 4 
the term which has p” ¢® for its argument in the expansion of (p + ¢)". tit 
. el) irae =1 
By the binomial theorem, this term is eed XP n 
Ex. 10. Thus, if n shillings are thrown into the air, in order to find what 
is the probability of any particular combination, it is only necessary to take 
: I Wi 
the corresponding term in the developement of 5 he ot : 
& ut 
Suppose there are five shillings : 
14 13% IN IN: Bid. B.4/1V 1) I'y 
l= A= — = — | = 5.(= = 0 
ta (5)+3 G+ 5G) +) + G+) . 
IN 
"The probability that they will all fall heads = (3) 
1\s 
4 heads and 1 tail = 5 x (5) 
1\s 
3 heads and 2 tails = 10 X (3) 
1\s 
2 heads and 3 tails = 10 x (;) 
: IY 
1 head and 4 tails = 5H % (3) 
5 
Hiolls s= (3) 
2 
18. The same theorem may be extended to any number of conflicting 
events, so that if P, Q, R, S represent any conflicting events of which the 
probabilities are p, ¢, 7, $, respectively, the probability of the event which is 
composed of @ times P, b times Q, c times R, d times S, &c.ina + b+¢ + d 
repeated trials is the term in the expansion of (pq --r+s,) “+2 +¢+2 which 
has p° ¢’ 7° ¢* for its argument; by the multinomial theorem this term is 
etd pot det otegd ~1), oo: 8.2.10, suid 
1. BiBe: iB Ze BrcesB TB 2, cl uB Be eal 1 r Ts