Correct solutions were submitted by Rich Bernstein.  Further correct solutions were received from Juan Marivela, Paul Botham, Jayavel Sounderpandian,  Al Zimmermann, Lou Cairoli, Juan Carrara, Nancy Schwarzkopf, Ron Welch.

The  probability distribution for the sum of two (fair) standard dice is 

There is only one other pair of dice that will yield this probability distribution with positive integer on the faces of the dice.  They are known as Sicherman dice and have the labelings 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8.  

This can either be determined by trial and error, or by using generating functions.  

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