ON PROBABILITY. 
' / ’ r. 
3 aT ar 6 a A= as, 
multiplying this by the probability of the hypothesis, the probability of 
drawing m/' white balls and 2/ black balls, considering only this hypothesis, is 
(m+n) +a -1)...... 1 
1.2.89, 77 .1,%.8.. = 
CAn)t™ (A —iapa)ytr 
3% Aa"(l— Az) +R@ara)"(1 - 2 Az) + &e’ 
and the probability of drawing m/ white halls and 2 black balls, considering 
all the hypotheses, is 
(n+ ny(on + w—1)....1 
1.2.8... 1.2.8. + 
> [= "tm (1 — A gy CAD" (1-2 A 2) 4 &e. 
Let A xz be infinitely diminished, and let: A 2 — 1, that is, let any ratio of 
white balls to the whole number of balls contained in the bag to A x be equally drs 
possible between zero and unity, then this expression becomes 
(m+n). (mm +n=-1)..... 1 i ad QA —-2)t” dz ba 
L.2. 8m .. 1.2.3. & Sd -2dz 
taicen between the limits + = 0 and 2 = 1. k 
aH 7 {hen 
Ja =a) ds = 1 (A — a) 4 TTY Sart (1 — a)=1d a bys 
¥ 1! 7 mo rer in 
EST (1-2) + GIDE Ine 2 (1 =a) 
i n.(n—1) m2 7 =~ 2 . h 
ET nl Tarde | 
ees Zh d= z)" + 2 gn (1 — zy" 1 prot 
m1 (m +1) (m2) 
n.(n—1) ss n= v 
TorD mtn mEy CT . 
ala de. wel Par. (n-2y....1% EL ™ 
(m~+1)(m+2).. m4+n24+1) : 
When x = 0, all these terms vanish ; when # = 1, all vanish except the last; P 
therefore the integral required, taken between the limits = 0 and wx = 1, is ; 
n(n —-Dm—-2)......1 
(m+ 1) (m +2) (m+ 3).. (mn +n +1) uy 
Yf ane (1 = 2)"*" d a, taken between the same limits, is 
(rda)Ym+2 —D(m+2—2)...... 1 
(m +m +1) (m+m'+2).... m+n +n+n +1) 
therefore the probability required is 
mtn) (m+a~1).....1 
1.2.3.9. 1.2.8... ®@ 
(ha) n 1-2 = 1) iveeanil 
X (m+m' +1) (m+ 2) .. (mm + no -+.1) 
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