(requesting math help) The value of a reroll.

I have long heard that reroll abilities are some of the most powerful effects in the Pathfinder system. While I can vaguely understand this concept, since it can be used to take an enemy save (often, even after the enemy makes the save originally) or help an ally retry a failed, I could never get a concrete idea of how effective it is.

I understand that you can determine the average roll of a particular dice. I can determine the average damage of a sneak attack or the average bonus from using inspiration. However, many reroll abilities are designed to skew the average- they tell you to take the better or worse roll. I am unsure how to boil that into something as direct as 'adds 3.5' like I do when I just add a 1d6.

As an example- I have been looking at medium builds, and I noticed how well the spell ill omen works with a melee medium. Since it is a no save spell that leads to multiple rerolls- so I barely need any CHA; even if it has an action to cancel it, that still takes up the enemy's time. The spell allows you to force the enemy to reroll/take worse roll multiple times- enough times that the spell could be used as a defense spell when the enemy gets close to a squishy caster (since the spell can affect enough d20 rolls to debuff an entire full attack).

But would it be effective as a defensive spell? If it only lowers the enemy's attack by -2 on average, then it is hardly worth it at higher levels. But if it lowers it by an average of -5, then that could allow the party wizard to avoid most of the full attack.

In summary- What is the average result of a 'reroll a d20 and take the worse result'? What is the average for a 'reroll and take the better result'?

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Without getting too far into the weeds with probability curves:

"Roll twice take the worst" — the median result is 6. So 50% of the time the result will be 6 or less.

"Roll twice take the best" — the median result is 15. So 50% of the time the result will be 15 or more.

If you did the math yourself:
Yes, it's actually 51% for both those numbers. I didn't want to confuse the issue.

Bear in mind that in Pathfinder (not Starfinder) most abilities are literal rerolls. See what's on the dice then decide whether or not to reroll it and take the second result regardless of which was better. (Not how ill omen works, but is for many abilities.) For abilities that use the second outcome, the average outcome is still 10.5. ("The dice have no memory.") They do however have the advantage of being more efficient since you usually decide to use them after seeing the first roll.

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You are kind of missing the point worrying about average rolls. What is important is the chance of failure or success. When you have to roll twice and take the worse results it means you have to succeed both rolls or you fail. Let’s say you need to roll a 11 or better, that is a 50% chance of succeeding. So your chance would be 25%. If you need a 19 or better that means you have 10% chance of success on a single roll. If you need to reroll and take the worse result you again have to make both rolls. So you have a 1 in 10 chance to make the first roll and if you make the first roll you have to do the same thing again. So that gives you a 1% chance of success.

The same is true for rerolling and taking the better result. If have a 50% chance to succeed and can reroll and take the better result I have a 75% chance to make it. If have a 10% chance of success and get a reroll and take the better result than my chance is now 19% instead of 10%.

If the reroll is you have to take the second roll no matter what the results that completely changes things.

Belafon wrote:

Without getting too far into the weeds with probability curves:

"Roll twice take the worst" — the median result is 6. So 50% of the time the result will be 6 or less.

"Roll twice take the best" — the median result is 15. So 50% of the time the result will be 15 or more.

** spoiler omitted **

Bear in mind that in Pathfinder (not Starfinder) most abilities are literal rerolls. See what's on the dice then decide whether or not to reroll it and take the second result regardless of which was better. (Not how ill omen works, but is for many abilities.) For abilities that use the second outcome, the average outcome is still 10.5. ("The dice have no memory.") They do however have the advantage of being more efficient since you usually decide to use them after seeing the first roll.

Yeah, I realize that the 'just take the second result, for better or worse' rerolls end up with normal averages. It was just hard to determine what happened when the averages got skewed by the take worse/better options.

So effects like ill omen do about a minus 4 or more debuff to the roll? (based off of how far it tends to take you from the average of 10.5) Cool. That puts it in the same area as evil eye from a high level witch. That is worth it both for debuffs on both saves and iterative attacks.

And Mysterious Stranger- I realize that the ability to spontaneously mess with an opponent's/ally's rolls outside of your turn (with immediate actions) is particularly powerful. Even though the luck domain uses the the 'take the second result, no matter what' are effective, since you can take a natural 20 or a natural 1 and turn it around. It takes a great or terrible roll, and then tends to bring the roll closer to 'average'.

But some reroll abilities are not quite that flexible. And I want to know the effect of the 'take better/take worse' part. I use Ill Omen as an example- you have to use an action during your own turn to apply the effect, and you have no control over what actions use up the rerolls. But as Belafon has shown, the effect of 'take the worse roll' is still strong enough to have a major debuff on the enemy.

(courtesy note- edited out a discussion on the qualities of a reroll; it cluttered up this discussion. I will also note that I added in various other things here and there while editing)

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Most reroll abilities in Pathfinder are intended for outcomes of "wow, that was a really bad roll, not much worse I can do on that". Most of them are used after the die is rolled but before the actual outcome is announced, so if you're pretty sure that the natural 3 you rolled isn't going to cut it even after your modifiers, you can reroll and try your luck again.

Bloodrealm wrote:
Most reroll abilities in Pathfinder are intended for outcomes of "wow, that was a really bad roll, not much worse I can do on that". Most of them are used after the die is rolled but before the actual outcome is announced, so if you're pretty sure that the natural 3 you rolled isn't going to cut it even after your modifiers, you can reroll and try your luck again.

I suppose a better way to summarize my question is "how much of an effect does take worse/take better have on rolls?". I wanted to know if it was worth it to learn and use ill omen on a regular basis, even though it isn't that kind of fantastic, flexible reroll ability.

And the effects seem to be enough to be worth it. At the very least, it can encourage enemies to spend the move action to remove the ill omen (so no full attacks or full round spells like summoning).

Mysterious Stranger wrote:
You are kind of missing the point worrying about average rolls. What is important is the chance of failure or success. ...

Mysterious Stranger's point is vital if you know what result will succeed. Suppose your character has to make a Will save, with +7 to the save and DC 20. That means rolling a 13 or higher, 8 chances out of 20. If the first roll is 14, then fine, don't reroll. If the first roll is 10, then rerolling is the only chance to prevent failure. The 8 out of 20 (40%) chance for success gains an extra (12/20)(8/20) = 96/400 = 24% chance of success for a total 64% chance of success.

That is the equivalent of an additional +5 on the Will save. But the harder the save, the less advantage. A DC of 22 on the same roll has a 30% chance of success on the first roll and a 21% chance of needed and succeeding at the reroll. That is closer to an additional +4 on the first roll.

The reroll gets trickier if we don't know what roll is necessary for success. A lot of reroll abilities say, "before the result is announced." The GM says, "Make a Will save," and does not tell you the DC. The question is reduced to, "Will the reroll be better?"

Well, if the first roll was a nat 1, then the reroll cannot be worse. On a nat 2, only a reroll of 1 would be worse. And so on. The problem comes up when the roll is 8 or higher. On a roll of 8, there are 7 chances that the reroll will be worse, 1 chance of another 7, and 12 chances the reroll will be better. If 8 was enough to succeed, then the reroll has a significant chance of turning success into failure. Does this balance the chance of changing failure into success? The odds of success getting a higher roll are better than the odds of getting a lower roll, but the advantage is pretty small, only +2.5. If the reroll were a once-per-day ability, I would not waste it on a mere +2.5.

As has been said it depends on the odds of success. Let's say an opponent has a 50% chance of making a save. If you cast the spell with persistent metamagic then the opponent will fail 75 percent of the time against the spell. Forced rerolls like persistent and misfortune are force multipliers.

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The following chart gives you the chances of making the roll. The first column is what you need to roll on a d20. The second column is the percent chance of making that roll. The third column is the chance to make the roll when under the effect of Ill Omen.

20 0.05 0.0025
19 0.1 0.01
18 0.15 0.0225
17 0.2 0.04
16 0.25 0.0625
15 0.3 0.09
14 0.35 0.1225
13 0.4 0.16
12 0.45 0.2025
11 0.5 0.25
10 0.55 0.3025
9 0.6 0.36
8 0.65 0.4225
7 0.7 0.49
6 0.75 0.5625
5 0.8 0.64
4 0.85 0.7225
3 0.9 0.81
2 0.95 0.9025
1 1 1

As you can your chance of actually making the number goes down pretty quickly. If you need a 2 or better to succeed the spell does not really do that much. In this case Ill Omen has less than a 5% reduction in your chance to succeed. But the chance to succeed quickly goes down from there. By the time you get to need a 7 Ill Omen is reducing the chance to succeed by 21%. One you get to need an 11 or better you have severely reduced the chance of your opponent succeeding. Beyond 14 you have dropped the chances of your opponent succeeding into the single digits.

Using a table like Mysterious Strangers' and converting the percentage differences to increments of 5% (i.e. points on a d20) you can see that a reroll varies in value from about 1 to 5 (+ or - depending on which kind of reroll). I usually assume plus or minus 3 when judging the effect.

As you can see from the other responses, the answer to a probability question completely depends on how you set up the question. Critically it depends on how much information you have. In this case determining what is the "best" choice depends on knowing what the target numbers are (what the enemy needs to roll). Which you usually won't.

lemeres wrote:
So effects like ill omen do about a minus 4 or more debuff to the roll? (based off of how far it tends to take you from the average of 10.5)

Yes and no. I gave you the median number. 50% of the time it will be 6 or less. So it's only a minus 4 half the time. It will still be 11 or above 25% of the time.

If you want to use a rule-of-thumb number for how much ill omen is worth you probably want to use the weighted mean, which is 7.125. So -3.375. Which rounds to -3.

edit: Mysterious Stranger's table is far more complete, but I get the sense you were just looking for a rough number.

If you have a caster with very high DC spells, ill omen is a dagger. It just depends on your party and if you can make it work.

Here is a graphical representation of what the OP is asking

The average on 2d20 take low is ~7.2
The average on 2d20 take high is ~13.8

However, it isn't a normal distribution so you can't just say +/-3.3, although that may be good enough in a pinch.

Create Mr. Pitt wrote:
If you have a caster with very high DC spells, ill omen is a dagger. It just depends on your party and if you can make it work.

Yeah. I was looking to make a champion focused medium that still used its spells to still remain relevant. And save debuffs are almost always relevant. Adding save debuffs on top of a well optimized DC makes it far more reliable.

If we are looking at a rule of thumb as around -3... yeah, great for save debuffs, but maybe not as great as a melee defense debuff. Well, it at least it pretty much negates any risk of crits (if they manage to threaten a crit twice then confirm it while under rerolls... then that monster DESERVES to eat our wizard).