[ 384 ] 
PROP. 7. 
If the probability of an event be a , and that of its 
failure be b in each fingle trial, the probability of its 
happening p times, and failing y times m p-\-q trials 
is E d b* if E be the coefficient of the term in which 
occurs d b ’ when the binomial a b\ b ^ rq is ex- 
panded. 
For the happening or failing of an event in differ- 
ent trials are fo many independent events. Where- 
fore (by cor. 2. prop. 6.) the probability that the event 
happens the iff trial, fails the 2d and 3d, and hap- 
pens the 4th, fails the 5th, &c. (thus happening and 
failing till the number of times it happens be p and 
the number it fails be q) is abb ab &c. till the 
number of as be p and the number of b's be y, that 
is; ’tis d b\ In like manner if you confider the event 
as happening p times and failing q times in any other 
particular order, the probability for it is d b 1 ; but 
the number of different orders according to which an 
event may happen or fail, fo as in all to happen p 
times and fail y, i \\p \ q trials is equal to the num- 
ber of permutations that aaaa bbb admit of when 
the number of as is p, and the number of b’s is y. 
And this number is equal to E, the coefficient of the 
term in which occurs a p b q when a -\-b\ pJtq is ex- 
panded. The event therefore may happen p times 
and fail y in p -j- y trials E different ways and no 
more, and its happening and failing thefe feveral dif- 
ferent ways are fo many inconfiftent events, the pro- 
bability for each of which is a p b q > and therefore by 
prop. 
