ON PROBABILITY. 
m+) (m+2).. m+n-+1) 
» 4 —————— fe 
7,7 —~1)......i1 
a Liha) AH - 1)... 
. KJ yaw 1 2 ETE 
% (+1) (+2)... (n+) (m+1) (m+2).. (m+n +1) 
(m+n+2)..(m+n+3)..(m+n+4)..m+ mw +n-Fn'+1) 
Fig If(n+1) (+2) .... (n+ n') berepresented by [n + 1] 
— Er m4-1)(m+2).... (mm) ” “ [m+ 17" 
jl and m~+n+2)(m+n+3)......(m+m' +n+n' + 1)by 
[m + n-- 2] +", this probability is expressed by 
m—+n)y(m +2 -1)....1 [n+1]"[m + 11" 
"4 ho 1.2.8....m ..1.2.8...00 [mn 27°1" 
iar Which result in this form may be easily remembered, by observing that 
tO of it is the same (with the difference of notation) as if the simple probability of 
any ratio g 
2 be equa) 1 21 : 
2 be equalls drawing a white ball was aL and the probability of drawing a black 
ball was gl. 
m+n + 2 
Ex. 18. Let us suppose the sun to have risen 2000000 times, or days, 
then the probability that it will rise again is given by the preceding formula, 
tf da by making p = 2000000, and g = 0, the probability required is £336091, 
This probability, which.is already very great, must be very considerably 
increased, if the discoveries of physical astronomy are taken into account. 
49. If a white ball has been drawn p times, and a black ball ¢ times, the 
ig; probability of drawing a white ball in a future trial is Tay and the 
: 3 +1 
robability of drawing a black ball is —1 —_ . the greater nd ¢ be- 
probability 8 prgt2’ > Pp and gq be 
(1-70 come, the more nearly do these fractions coincide with —2— and il 
p+ gq PTq 
which are their limits when p and g are indefinitely increased. This theorem 
is the converse of Bernoulli's theorem, of which we gave a demonstration, 
p- 21. 
ot the last; Ex. 19. We said, that if a shilling was tossed into the air, the probability 
z=Lis ofits falling heads, or the reverse, twice running, was rather greater than i. 
Let the probability of the shilling falling heads. be any of the following 
quantities : 
poo 1 : 1 11 1 , yer) 
5 — mt en Dax 5 =z 23TH 3 +@@—-1Da, Bt TL 
The number of hypotheses is 2 7 + 1, and the probability of the shilling 
falling heads twice running is 
| 2 1 2 1 2 1\2 
(G-i2) +{5-G-D4 ee +f =f +(5)] 
2i41 I. % 11 : 3 1 2 
+(3+ic) + {3+ G—1sf coven oF {5 +0) 
99