May 26th, 2017, 08:04 AM  #1 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,817 Thanks: 760  Coupla pairs
Skeeter has 4 decks of cards, each labelled 1 to 13. He puts 'em all together and shuffles 'em generously. Then he deals 2 cards to Archie. Following that, he deals 2 cards to Joppy. What be the probability that both Archie and Joppy did NOT get a pair? 
May 26th, 2017, 05:34 PM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,817 Thanks: 760  
May 26th, 2017, 05:46 PM  #3 
Senior Member Joined: Aug 2012 Posts: 1,765 Thanks: 480  
May 26th, 2017, 05:54 PM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,817 Thanks: 760 
Archie did not get a pair. Joppy did not get a pair. 
May 26th, 2017, 06:29 PM  #5 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,533 Thanks: 513 Math Focus: Yet to find out. 
Knowing Denis, the solution will be a coupla cute numbers.

May 26th, 2017, 06:39 PM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,817 Thanks: 760  
May 29th, 2017, 01:28 PM  #7 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 590 Thanks: 81 
Did I do this right? There are 6 possible pairs for each number, which is 78 total. There are 52 C 2 = 1,326 ways to be given 2 cards. The probability Archie got a pair is 78/1,326 = 1/17, and the probability he didn't get a pair is 16/17. At that point, 11 numbers that Archie didn't get have 6 possible pairs, and the 2 numbers Archie got have 3 possible pairs, for a total of 72. There are 50 C 2 = 1,225 ways to be given 2 of the remaining cards. Joppy has a 72/1,225 chance of getting a pair and a 1,153/1,225 chance of not getting a pair. The probability of neither person getting a pair is (16/17)(1,153/1,225) = 18,448/20,825 = about .8859. 
May 29th, 2017, 03:34 PM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,817 Thanks: 760 
Thanks Evan. Given answer, which I didn't agree with: 48/51 * 46/49 = ~.88355 I ran 2 simulations of 1 million each; got 885,781 and 885, 587 You've confirmed my results... 
May 30th, 2017, 05:42 AM  #9 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Hey , how come I don't get to play 
May 30th, 2017, 05:49 AM  #10 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,533 Thanks: 513 Math Focus: Yet to find out.  

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