samplespace=Flatten[Table[{a,b,c}, {a, 1,6}, {b,1,6}, {c,1,6}],2]

Count[samplespace, x_/;EvenQ[Plus@@x]]

Table[
Count[samplespace, x_/;And[IntegerQ[(Plus@@x)/2], x[[1]]==j]], {j,1,6}]

Table[Count[samplespace, x_/;And[IntegerQ[(1+Plus@@x)/2], x[[1]]==j]], {j,1,6}]

Table[Count[samplespace, x_/;And[IntegerQ[(Plus@@x)/3], x[[1]]==j]], {j,1,6}]

Table[Count[samplespace, x_/;And[IntegerQ[(1+Plus@@x)/3], x[[1]]==j]], {j,1,6}]

Table[Count[samplespace, x_/;And[IntegerQ[(2+Plus@@x)/3], x[[1]]==j]], {j,1,6}]

(*****************************************)

Simplify[1-(1-p)(1-q)]

(*Problem 3.5:10*)

Module[{x=1-(5/6)^4}, {x, N[x]}]

Module[{x=1-(35/36)^24}, {x, N[x]}]

(*Extension 3.5:10*)

Module[{x=4*(1/6)(5/6)^3}, {x, N[x]}]

Module[{x=24*(1/36)(35/36)^23}, {x, N[x]}]

Module[
{f,s},
f[2]=1/36;f[3]=2/36;f[4]=3/36;f[5]=4/36;f[6]=5/36;f[7]=6/36;f[8]=5/36;f[9]=4/36;f[10]=3/36;f[11]=2/36;f[12]=1/36;s=Sum[(1-f[k])^5f[k], {k, 2, 12}];
{s, N[s]}
]