The two dice are tossed. What is the probably of the event that the sum of two numbers on the both dice at least equal 6 and at most equal 9?

With out loss of generality we can assume one die is #color(red)("red")# and the second die is #color(green)("green")#

For each of the #color(red)(6)# faces on the #color(red)("red die")# there are color(green)(6) different possible outcomes on the #color(green)("green die")#.
#rArr# there are #color(red)(6) xx color(green)(6) = color(blue)(36)# possible combined outcomes.

Of these outcomes

A total of 6 can be achieved in #color(cyan)(5)# ways:#{(color(red)(1),color(green)(5)),(color(red)(2),color(green)(4)),(color(red)(3),color(green)(3)),(color(red)(4),color(green)(2)),(color(red)(5),color(green)(1))}#

A total of 7 can be achieved in #color(cyan)(6)# ways:#{(color(red)(1),color(green)(6)),(color(red)(2),color(green)(5)),(color(red)(3),color(green)(4)),(color(red)(4),color(green)(3)),(color(red)(5),color(green)(2)),(color(red)(6),color(green)(1))}#

A total of 8 can be achieved in #color(cyan)(5)# ways:#{(color(red)(2),color(green)(6)),(color(red)(3),color(green)(5)),(color(red)(4),color(green)(4)),(color(red)(5),color(green)(3)),(color(red)(6),color(green)(2))}#

A total of 9 can be achieved in #color(cyan)(4)# ways:#{(color(red)(3),color(green)(6)),(color(red)(4),color(green)(5)),(color(red)(5),color(green)(4)),(color(red)(6),color(green)(3))}#

Since these event are mutually exclusive there are
#color(white)("XXX")color(cyan)(5+6+5+4) =color(brown)(20)# ways of achieving #{6,7,8,9}#

So the probability of achieving #in{6,7,8,9}#is
#color(white)("XXX")color(brown)(20)/color(blue)(36)= 4/9#