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The availability of a Charm/Intimidate option depends on the vanilla threshold, and the number of Paragon/Renegade points adjusted for bonuses (Persuasion points):
Chance of a Persuasion option being available = Base% + 0.4 x ( sqrt(Persuasion points / Vanilla Threshold) -1).
And there is a chance to get a critical success (option available no matter the threshold/persuasion points), or a critical fail (option unavailable no matter the threshold/persuasion points).
If Persuasion Points = Vanilla Threshold, the chance of the option being available is equal to the Base%.
If Persuasion Points > Vanilla Threshold, the chance increases. And if Persuasion Points < Vanilla Threshold, the chance decreases. The sqrt is present in the formula to reduce the impact of large differences, which are actually rather easy to obtain (it's possible to have 2x or 3x more Persuasion Points than the Vanilla Threshold).
[Easy] Base% = 60%, critical success = 30%, critical fail = 10%.
[Normal] Base%. = 50%, critical success = 20%, critical fail = 20%.
[Hard] Base% = 40%, critical success = 10%, critical fail = 30%.
If a single conversation has mutliple Persuasion checks of increasing difficulty, there is only 1 random event, which will determine how far you can go (if you pass the hardest check, you automatically pass the easier ones as well). Otherwise, it would make such checks much more difficult because of the need to succeed several times in a row.
3) Examples
Example 1. In vanilla, the Miranda/Jack conflict has a Charm option with Max difficulty, and an Intimidate option with Hard difficulty. The Vanilla Thresholds will depend on when this happens during your playthrough, but in mine they were at 1363 for Charm and 693 for Intimidate. On a mostly renegade playthrough, I had 558 Charm points (Paragon points adjusted for bonuses) and 1975 Intimidate points. With the [Normal] difficulty option, this means 35% chance of having the Charm option available, and 78% chance of having the Intimidate option available (and thus 86% of having at least one of the two options available). With [Easy], it would be 45% and 88% (93% of having at least one). With [Hard], it would be 25% and 68% (76% of having at least one).
Example 2. The same conflict, at the same time in a Paragon playthrough with 1975 Charm points and 558 Intimidate points, would give in [Normal] 58% to have the charm option available, and 46% to have the Intimidate option available (77% to have at least one). In a paragade/renegon playthrough with 1266 Charm points and 1266 Intimidate points, it would give in [Normal] 49% to have the charm option available, and 64% to have the Intimidate option available (82% to have at least one).