418 Proceedings of the Royal Society 
As a final example we have the singular for mula, 
1 
x-y 
y 
x(x + i ) 
+ 
y(y+i) 
x(x + !)(# + 2) 
+ &c. 
whence it follows that, subject to the introduction of the remain- 
ders as above (which vanish if the series are extended to infinity, 
and if x > y ), 
1 , v . y(y+ 1) . V 1 y , >Av- v ) . 
x x(x+\) x(x+ l)(a?-f- 2) x(x+\) x(x+l)(x + 2) 
1 . y 2 y 2 (y 2 +^) . 
x 2 x 2 (x 2 + 1 ) x\x 2 + 1 ){x 2 + 2) 
By another application of the formula we may easily obtain finite 
expressions for the sum of the series of which two successive terms 
are 
y( y+ 1 ) • • • ♦ (y + r- 1) , y(y+ 1) .... ( y + r ) 
x(x+ 1) . . . . (x + s - 1) * ‘ x(x+ 1) .... (x + s) * 
I obtained the first expression above by integrating by parts a 
power such as xP~ l , but the following mode of obtaining it shows at 
once its nature. 
Let there be a number of independent events, A, B, . . . . N, 
whose separate probabilities are a, (3 , ... v. Then the chance that 
one at least of them occurs is 
1 — (1 — a)(l — /?) . . . . (1 — v) . 
But we may obtain another expression for the same result by writ- 
ing the chance that any one (say A) occurs, adding to that the 
chance that another (say B) occurs while A does not occur, then 
that C occurs and neither A nor B, &c. This gives 
a + — a) + y(l — a)( 1 
Equating these two expressions we get an identity which is easily 
transformed into that first given. 
But its truth is much more easily - seen if we write a for (1 - a), 
&c., when the last given form becomes 
1 - a/3'y'. . . . = 1 - a' + (1 - /3')a + (1 - yV/T + . . . . 
