Count the ordered roll pairs producing each value. The probability mass functions are
(1)
and are zero outside these ranges. Both counts have a unique peak:
(2)
However, forces equal rolls, whose sum is even. Hence , whereas the product of the two marginal probabilities is . This disproves independence of random variables:
(3)
Figure 1.
The 36 possible sum and difference pairs for two independent fair dice; the two marginal modes cannot occur together
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