Now let's instead assume the proc is a DD that does 44 damage.
1) If both swings proc, the total damage is 88
2) If just the first swing procs, the total damage is 44.
3) If just the second swing procs, the total damage is 44.
4) If neither swing procs, the total damage is 0.
Case 1 happens with probability 1/12 * 1/12
Case 2 happens with probability 1/12 * 11/12
Case 3 happens with probability 11/12 * 1/12
Case 4 happens with probability 11/12 * 11/12
The total expected damage is the sum of the expected damage of each of the four cases, weighted by the probability
88 * 1/144 + 44 * 11/144 + 44 * 11/144 + 0 * 121/144
The total expected damage is 7.33
Your calculation in this case would be: on average, there's a 1/6 chance of a proc. 1/6 * 44 is 7.33. The numbers match, because the damage is linear to the number of procs.
It does not match in the DoT example, because the damage is not linear to the number of procs
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