Let, #E# be the event the sum on two number (no.) cubes is even,
and, #F#, the sum be a multiple of #3#.
#"The Reqd. Prob."=P(EuuF)=P(E)+P(F)-P(EnnF)#.
By the order pair #(x,y)#, we wish to denote that the no. #x# and #y#
have come up on the face of the #1^(st) and 2^(nd)# no. cube, resp.
Clearly, the sample space #S# associated with the given random experiment is given by,
#S={(1,1),(1,2),...,(1,6),(2,1),(2,2),...,(2,6),...,(6,1),(6,2),...(6,6)}.#
#:."The no. of elements in "S=n(S)=36#.
Further,
#E={(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5),(4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4),(6,6)},#
#F={(1,2),(1,5),(2,1),(2,4),(3,3),(3,6),(4,2),(4,5),(5,1),(5,4),(6,3),(6,6)},#
#EnnF={(1,5),(2,4),(3,3),(4,2),(5,1),(6,6)}#.
#:. n(E)=18, n(F)=12, n(EnnF)=6#.
#:. P(E)=(n(E))/(n(S))=18/36, P(F)=12/36, P(EnnF)=6/36#.
#:."The Reqd. Prob."=18/36+12/36-6/36=24/36=2/3#.
Enjoy Maths.!
