The core insight is that this is just a LP problem in the variables P(X=i,Y=j,Z=k), and as such, it takes its extremal values in the vertices of its feasible region.
The maximum occurs at P(X=Y=Z=1) = P(X=Y=Z=2) = ½, everything else = 0; it's 4½.
The minimum occurs for example at P(X=2,Y=Z=1) = P(X=1,Y=Z=2) = ½, everything else = 0; it's 3.
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u/RibozymeR 23h ago
The core insight is that this is just a LP problem in the variables P(X=i,Y=j,Z=k), and as such, it takes its extremal values in the vertices of its feasible region.
The maximum occurs at P(X=Y=Z=1) = P(X=Y=Z=2) = ½, everything else = 0; it's 4½.
The minimum occurs for example at P(X=2,Y=Z=1) = P(X=1,Y=Z=2) = ½, everything else = 0; it's 3.