﻿AND 
  ITS 
  APPLICATION 
  TO 
  A 
  PROBLEM 
  IN 
  PROBABILITIES. 
  543 
  

  

  All 
  that 
  we 
  can 
  say 
  is, 
  that 
  experience 
  has 
  led 
  dispassionate 
  men 
  to 
  come 
  to 
  nearly 
  

   the 
  same 
  conclusion 
  as 
  the 
  mathematician 
  : 
  for 
  while 
  he 
  asserts 
  the 
  probability 
  

  

  of 
  success 
  to 
  be 
  *_ 
  2 
  , 
  they 
  act 
  upon 
  the 
  supposition 
  that 
  the 
  probabilities 
  of 
  

  

  success 
  and 
  failure 
  are 
  proportioned 
  to 
  the 
  number 
  of 
  experienced 
  cases 
  of 
  success 
  

   and 
  failure 
  : 
  and 
  when 
  either 
  ^> 
  or 
  q 
  is 
  a 
  large 
  number, 
  that 
  is, 
  when 
  the 
  experience 
  

   is 
  great, 
  the 
  conclusion 
  and 
  the 
  supposition 
  coincide. 
  

  

  In 
  order 
  to 
  realise 
  the 
  Problem, 
  we 
  shall 
  use 
  the 
  ordinary 
  illustration, 
  and 
  

   suppose 
  that 
  a 
  bag 
  contains 
  m 
  balls 
  in 
  unknown 
  proportions 
  of 
  black 
  and 
  white, 
  

   but 
  all 
  either 
  black 
  or 
  white; 
  that^> 
  white 
  and 
  q 
  black 
  balls 
  have 
  been 
  drawn, 
  

   and 
  that 
  it 
  is 
  required 
  to 
  find 
  the 
  probability 
  of 
  drawing 
  a 
  white 
  at 
  the 
  p 
  + 
  q 
  + 
  v** 
  

   drawing. 
  

  

  The 
  Problem 
  as 
  thus 
  stated, 
  admits 
  of 
  four 
  varieties. 
  

  

  1. 
  m 
  may 
  be 
  given, 
  and 
  the 
  balls 
  drawn 
  may 
  have 
  been 
  replaced 
  in 
  the 
  bag. 
  

  

  2. 
  m 
  may 
  be 
  given, 
  and 
  the 
  balls 
  drawn 
  not 
  replaced. 
  

  

  3. 
  m 
  may 
  be 
  infinite 
  or 
  indefinite, 
  and 
  the 
  balls 
  replaced. 
  

  

  4. 
  m 
  may 
  be 
  infinite 
  or 
  indefinite, 
  and 
  the 
  balls 
  not 
  replaced. 
  

  

  Of 
  these, 
  the 
  3d 
  is 
  the 
  only 
  case 
  which 
  I 
  find 
  solved 
  in 
  the 
  treatises 
  which 
  I 
  

   have 
  consulted. 
  I 
  propose 
  to 
  solve 
  the 
  2d 
  case, 
  and 
  therein 
  the 
  4th 
  ; 
  and, 
  in 
  

   conclusion, 
  to 
  make 
  an 
  attempt 
  at 
  the 
  solution 
  of 
  the 
  1st 
  case. 
  

  

  To 
  render 
  the 
  observed 
  event, 
  that 
  is, 
  the 
  drawing 
  of 
  p 
  white 
  and 
  q 
  black 
  

   balls 
  (or 
  E), 
  possible, 
  the 
  original 
  number 
  of 
  whites 
  may 
  have 
  been 
  any 
  number 
  

   from 
  m—q 
  to 
  p 
  inclusive, 
  and 
  the 
  number 
  of 
  blacks 
  any 
  number 
  from 
  q 
  to 
  m~p. 
  

  

  Let 
  us 
  call 
  the 
  hypothesis 
  of 
  m—q 
  white 
  and 
  q 
  black, 
  Hj 
  

   and 
  m—q—1 
  white 
  and 
  q 
  + 
  l 
  black, 
  H 
  2 
  , 
  &c. 
  

  

  „,, 
  TT 
  . 
  r, 
  , 
  , 
  .-.., 
  „-!-, 
  m 
  — 
  q 
  .m 
  — 
  q 
  — 
  1 
  .... 
  m 
  — 
  q 
  — 
  p 
  + 
  1x1 
  .2 
  .3 
  .... 
  q* 
  

  

  ThenH, 
  gives 
  for 
  probability 
  of 
  E 
  — 
  *— 
  — 
  * 
  , 
  i 
  -^ 
  

  

  10 
  r 
  J 
  m 
  . 
  m 
  — 
  l 
  . 
  ... 
  m 
  — 
  q 
  —p 
  + 
  1 
  

  

  or, 
  calling 
  the 
  denominator 
  A, 
  

  

  H 
  gives 
  -r 
  'tn 
  — 
  q.m 
  — 
  q 
  — 
  1 
  m 
  — 
  q— 
  p 
  + 
  1 
  x 
  1 
  . 
  2 
  . 
  3 
  q 
  (a) 
  

  

  A. 
  

  

  H, 
  gives 
  -T- 
  • 
  m 
  — 
  q 
  — 
  1 
  .m 
  — 
  q 
  — 
  2 
  .... 
  m 
  — 
  q—p 
  x 
  2 
  . 
  3 
  . 
  4 
  . 
  . 
  . 
  . 
  q 
  + 
  1 
  (/?) 
  ? 
  (F) 
  

  

  A 
  

   1^ 
  

  

  L 
  3 
  6"^^ 
  

  

  H 
  ., 
  gives 
  -r- 
  • 
  m, 
  — 
  q 
  — 
  2.m 
  — 
  q 
  — 
  3 
  .... 
  m 
  — 
  q—p— 
  1x3.4 
  .... 
  q 
  + 
  2 
  (y) 
  

  

  &C. 
  &C. 
  

  

  Now, 
  a 
  + 
  /3 
  + 
  7, 
  &c. 
  by 
  the 
  former 
  proposition 
  (E) 
  

  

  1 
  a 
  . 
  Q 
  — 
  l 
  1 
  -, 
  , 
  

  

  =. 
  -r- 
  . 
  * 
  w 
  , 
  h 
  i 
  xm 
  + 
  l.m 
  . 
  . 
  . 
  . 
  m—p—q 
  + 
  1 
  

  

  A 
  p 
  + 
  l,p 
  + 
  2. 
  . 
  . 
  p 
  + 
  q 
  + 
  1 
  

  

  •'■ 
  P 
  robabilit 
  y 
  ofH 
  i=^9^yT^. 
  

  

  * 
  The 
  coefficient 
  (U 
  of 
  Galloway's 
  Treatise), 
  expressing 
  the 
  number 
  of 
  different 
  ways 
  in 
  which 
  

  

  p 
  white 
  and 
  q 
  black 
  balls 
  can 
  be 
  combined 
  in 
  p 
  + 
  q 
  trials, 
  is 
  here 
  omitted. 
  This 
  is 
  immaterial, 
  

  

  a 
  

   as 
  it 
  disappears 
  in 
  the 
  expression 
  -^-. 
  -r— 
  

  

  