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In other words dice rolls are independent events and e.g. the chance to get two heads or two tails with a fair coin (dice = [["H", "T"], ["H", "T"]]) are \$P(HH \vee TT) = 0.5\cdot0.5 + 0.5\cdot0.5 = 0.5\$.

In other words dice rolls are independent events and e.g. the chance to get two heads or two tails with a fair coin (dice = [["H", "T"], ["H", "T"]]) are \$P(HH \vee TT) = P_1(H)\cdot P_2(H) + P_1(T) \cdot P_2(T) = 0.5\cdot0.5 + 0.5\cdot0.5 = 0.5\$.

\$P(overlap) = \sum\limits_j \prod\limits_i p^i_j = \sum\limits_j \prod\limits_i n^i_j / k_i\$,

$$ P(overlap) = \sum\limits_j \prod\limits_i p^i_j = \sum\limits_j \prod\limits_i n^i_j / k_i, $$

\$P(overlap) = \sum\limits_j \prod\limits_i p^i_j\$,

\$P(overlap) = \sum\limits_j \prod\limits_i p^i_j = \sum\limits_j \prod\limits_i n^i_j / k_i\$,