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Odds are ... (Posted on 2017-02-14) Difficulty: 3 of 5
You face an urn with 5555 cards in it, each has a non-zero integer written on it. Nothing is said about the distribution of those numbers. You are told to draw randomly a card, copy the number, return it back, shuffle and draw randomly a card, then write down the sum of both numbers, say S.

(i) Prove: The probability of S being an even number is higher than S being odd.
(ii) Is it true for any initial number of cards? Comment.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
probabilities | Comment 4 of 8 |
The probabilities from 1 to 10 cards are:

 1 	 1 	         1.0 
 2 3//4 0.75 
 3 2//3 0.6666666666666666666 
 4 5//8 0.625 
 5 3//5 0.6 
 6 7//12 0.5833333333333333333 
 7 4//7 0.5714285714285714285 
 8 9//16 0.5625 
 9 5//9 0.5555555555555555555 
 10 11//20 0.55 

Unfortunately there's an overflow for such large numbers as 5555 even in UBASIC. We could work around this using logarithms, but the point has been made.



from

   5    kill "oddsare.txt"
  10    open "oddsare.txt" for output as #2   
  20      for n=1 to 10  
  30       tot=0
  40       for i=0 to int(n/2)
  45         if i<n//2 then
  50          :condp=(1//2)^n * combi(n,i)*2
  55         :else
  56          :condp=(1//2)^n * combi(n,i)
  70         p=condp*((i//n)^2+((n-i)//n)^2)
  80         tot=tot+p
  90       next
 150       print n,tot,tot/1
 160       print #2, n,tot,tot/1
 170      next n   
 180    close #2  

  Posted by Charlie on 2017-02-14 11:34:26
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