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 January 26th, 2015, 04:59 AM #1 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 660 Thanks: 87 Probability Of Consecutive Numbers When Numbers Are Paired The numbers 1 to 20 are randomly made into ten pairs with every number used once. What is the probability that at least four of the pairs will have consecutive numbers? Note that most numbers can be consecutive with a number 1 higher or 1 lower but 1 and 20 are exceptions. I did 10 random number simulations and the most pairs with consecutive numbers I got was 2 pairs in one simulation. It only happened 9 times in the 10 simulations combined. Therefore I'm expecting a very low probability. January 26th, 2015, 05:32 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra How about looking at this method of generating arrangements? Let case $(n,p)$ be the problem of getting $p$ or more pairs out of $n$ consecutive numbers placed in $n$ positions. Case $(n,p)=(20,4)$: Place the 1 in one of the 20 positions. Now place the 2 in one of the 19 remaining positions. If it is not paired with the 1, you can no longer pair the 1, so we are now reduced to the case $(19,4)$ (with the first number already placed). (Note that this case has non-pairable places, but I'm not sure that it matters. If it is paired with the 1, we have one pair and we are reduced to the case $(18,3)$. Tags consecutive, numbers, paired, probability Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post EvanJ Advanced Statistics 7 March 8th, 2014 08:33 PM johnr Number Theory 5 March 5th, 2014 11:03 AM ganeshsrivatsav Number Theory 2 November 15th, 2013 08:43 PM daigo Algebra 1 May 18th, 2012 02:59 PM coax Number Theory 1 July 24th, 2009 05:36 AM

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