
Advanced Statistics Advanced Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 26th, 2015, 04:59 AM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 608 Thanks: 82  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: 7,385 Thanks: 2476 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 nonpairable 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Consecutive Numbers On A Die Probability Question  EvanJ  Advanced Statistics  7  March 8th, 2014 08:33 PM 
The Sum of Two Consecutive Tetrahedral Numbers ...  johnr  Number Theory  5  March 5th, 2014 11:03 AM 
Missing Numbers from a consecutive List.  ganeshsrivatsav  Number Theory  2  November 15th, 2013 08:43 PM 
How to add consecutive imaginary numbers  daigo  Algebra  1  May 18th, 2012 02:59 PM 
Two consecutive numbers  coax  Number Theory  1  July 24th, 2009 05:36 AM 