The d20 mechanic boils down to: 1d20+M >= TN, where TN = target number (e.g., AC, save DC) and M = total modifier relevant to the task. Equivalently, 1d20 >= TN-M.
Note that TN-M < 1 is the condition for auto-success, while TN-M > 20 is the condition for auto-fail; in other words, if the...