The totol number ofpossible pairs (i.e the sample set ) :> 2010C2 *2008C2 * 2006C2 ..... 4C2 * 1
If one pair has to intersect all the other intervals (I assume it is (1,2010) ), we can have 1004 other distinct pairs , this can be done in : 2008C2 * 2006C2 .... 4C2 *1
The probability is => 2008C2 * 2006C2 .... 4C2 *1 / 2010C2 *2008C2 * 2006C2 ..... 4C2 * 1 => 1/2010c2 = 2/ 2010*2009 = 1/(1005*2009)