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January 19th, 2010, 06:32 AM  #1 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Rule of succession problem
Taken from an example in the 1870 version of Todhunter algebra. I spotted and answered this problem (correctly?) on Yahoo answers.... You have a bag of 5 equally sized balls the colour of which you have no information about. You draw two balls and both are white. What is the probability that all five of the balls are white ? I answered the problem without resorting to the succession rule.....how would you solve it ? 
January 19th, 2010, 07:11 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Rule of succession problem
I have no idea how to solve the problem. I suppose that any solution must infer a reasonable Bayesian prior...? Link: Algebra for the Use of Colleges and Schools: With Numerous Examples, p. 466 ff. 
January 19th, 2010, 10:36 AM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Rule of succession problem Hello, CarlPierce! CRGreathouse is correct . . . This requires Bayes' Theorem. Quote:
We know there are at least 2 Whites among the 5 balls. I assume that the assortment colors of the balls are equally likely. [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color]  
January 19th, 2010, 12:51 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Rule of succession problem Quote:
 
January 21st, 2010, 07:37 AM  #5 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Re: Rule of succession problem
I solved it as follows Given 2 whites then the actual number of whites in the bag can be either 2,3,4 or 5 For each possible case the chance of drawing two whites is ... 2/20 , 6/20, 12/20 or 1 So the probability that the actual number of whites is 5 is therefore 1 / (2/20 + 6/20 + 12/20 + 1) = 0.5 The succession rule gives the same answer. namely 3/4 x 4/5 x 5/6 = 0.5 Is my reasoning valid ? Todhunters gives the answer 0.5 
January 21st, 2010, 08:07 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Rule of succession problem Quote:
First, it seems to rest on the assumption that, a priori, a ball has even odds to be white or nonwhite. I question this. But given that assumption, I would expect 1/32 prior odds that all 5 are white, not 1/4 or 1/6. This would give 1/(1*1 + 5*3/5 + 10*3/10 + 10*1/10) = 1/8 rather than 1/2. Laplace's "rule of succession" would suggest that the chance that a random ball is white is (2 + 1) / (2 + 2) = 3/4, making the chance that the three unknown balls are white 3/4 * 3/4 * 3/4. The same analysis could of course be applied here: # white balls (prior probability) 5 (3/4)^5 * (1/4)^0 * 1 = 243/1024 4 (3/4)^4 * (1/4)^1 * 5 = 405/1024 3 (3/4)^3 * (1/4)^2 * 10 = 270/1024 2 (3/4)^2 * (1/4)^3 * 10 = 90/1024 1 (3/4)^1 * (1/4)^4 * 5 = 15/1024 0 (3/4)^0 * (1/4)^5 * 1 = 1/1024 243/(243*1 + 405*3/5 + 270*3/10 + 90*1/10) = 27/64 The answers are the same either way. I tend to think that 27/64 is more reasonable than 1/8, but I certainly wouldn't go any higher than 27/64. In particular I'm not willing to go up to the Todhunter 32/64. The method of Agresti & Coull seems to suggest a yet lower figure: 8/27. The ClopperPearson 95% interval is maddeningly wide: 1.118% to 100%.  
January 21st, 2010, 08:27 AM  #7 
Newbie Joined: May 2009 Posts: 25 Thanks: 0  Re: Rule of succession problem
I'm not sure I need to assume anything about the prior probabilities of a ball being white. I simple state that if 2 white balls have been drawn then the respective changes it happened given the four possible states were 2/20, 6/20, 12/20 or 1 Can't we turn the odds around to infer the chance of each state actually being true. So is it not reasonable to infer that the P(white = 5) is the ratio of these ? i.e 20/(2+6+12+20) = 0.5 
January 21st, 2010, 08:36 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Rule of succession problem Quote:
Quote:
2/20, 6/20, 12/20 or 1 as you describe, but the only reasonable answer would be 1. Suppose instead that you knew the bunny left three kinds of bags: one with 5 white, one with 3 white, and one with 1 white; and that the respective probabilities were 1/4, 1/2, and 1/4. Then you would be justified in deriving a probability of 5/8. Quote:
 
January 21st, 2010, 09:40 AM  #9  
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Rule of succession problem Quote:
Yesterday was a weekday, and today is a weekday. What is the probability the next three days will be weekdays?  
January 21st, 2010, 12:12 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Rule of succession problem Quote:
 

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