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PROP. 6. 
The probability that feveral independent events 
lhall all happen is a ratio compounded of the proba- 
bilities of each. 
For from the nature of independent events, the 
probability that any one happens is not altered by the 
happening or failing of any of the reft, and confe- 
quently the probability that the 2d event happens on 
iuppofttion the ift does is the fame with its original 
probability ; but the probability that any two events 
happen is a ratio compounded of the probability of the 
1 ft event, and the probability of the 2d on fuppofition 
the 1 ft happens by prop. 3. Wherefore the probability 
that any two independent events both happen is a ra- 
tio compounded of the probability of the ift and the 
probability of the 2d. And in like manner confidering 
the ift and 2d event together as one event ; the proba- 
bility that three independent events all happen is a ratio 
compounded of the probability that the two ift both 
happen and the probability of the 3d. And thus you 
be accompanied with another to be determined at the fame time ? 
In this cafe, as one of the events is given, nothing can be due 
for the expectation of it ; and, confequently, the value of an ex- 
pectation depending on the happening of both events muff be the 
fame with the value of an expectation depending on the happen- 
ing of one of them. In other words ; the probability that, when 
one of two events happens, the other will, is the fame with the 
probability of this other. Call x then the probability of this 
h • P 
other, and if - be the probability of the given event, and — 
the probability of both, becaufe ~ ^ x x zz ^ zz the pro- 
bability mentioned in thefe proportions. 
may 
