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I think the disconnect here isn't about how the math works, but in appetite for success rate vs character investment.
If you use the Search activity you get only one chance of finding a trap before triggering it. Since most traps will be within a few levels of your own, the odds of success are somewhere in the 40% to 80%.
Deriven however feels that if you invested heavily in it, you should be almost certain to find the trap when looking for it. If you roll multiple times, it basically comes down to the chance that you'll roll a 1/critfail before you roll a success. Overall that gives far better odds than if you only rolled once.
I don't think this is how the game is intended. Traps are bad enough that if you're allowed to improve your odds this way, you should always do that, and the Search activity is so much worse that you should never use it.
Also, I don't think PF2 is designed with the idea that you can get so good at something that it'll almost always succeed. If you have to roll for something, there should be a decent chance of success and a decent chance of failure. By rolling multiple times you undermine that design.
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Also, if you can retry Seek indefinitely, then with one feat, Ageless Patience you basically "solved" all non-combat Seek problems, because you can never critically fail.

Witch of Miracles |
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It's clearly built into the game that you can repeat looking for things. Seeking the same area repeatedly in combat, trying to find an invisible enemy for your team? Plenty acceptable. No rules against it if you're convinced they're where you're looking. If players want to waste time on self-invented red herrings, you can let them.
Likewise, if someone truly suspicious of hazards wants to search for them forever, I personally don't see why they can't. I mainly see two outs for this:
1) Hazard checks are secret and you don't even know when they're being rolled. So you can't actively search for hazards, period. The GM rolls once, and you find them or you don't.
I don't like this very much. It just feels bad to not be able to be overly cautious. It does remove the problem and speed play, though. Some tables would prefer this.
2) Run the game in a way where the time cost is significant.
This is harder, but imo more rewarding for everyone involved. You give the players a choice with actual tradeoffs instead of making hazards uninteractive gotchas. However, when I say it's harder... it's a lot harder. A majority of scenarios are not written for time pressure to matter.
My players help with this one, because they typically feel a sense of urgency if the plot implies one. If people are kidnapped, they're already hesitant to spend a lot of time taking 10 minute breaks to heal. So naturally, they're hesitant to spend a lot of time combing the hallway for traps as well.

NorrKnekten |
It's clearly built into the game that you can repeat looking for things. Seeking the same area repeatedly in combat, trying to find an invisible enemy for your team? Plenty acceptable. No rules against it if you're convinced they're where you're looking. If players want to waste time on self-invented red herrings, you can let them.
Likewise, if someone truly suspicious of hazards wants to search for them forever, I personally don't see why they can't. I mainly see two outs for this:
1) Hazard checks are secret and you don't even know when they're being rolled. So you can't actively search for hazards, period. The GM rolls once, and you find them or you don't.
I don't like this very much. It just feels bad to not be able to be overly cautious. It does remove the problem and speed play, though. Some tables would prefer this.
2) Run the game in a way where the time cost is significant.
This is harder, but imo more rewarding for everyone involved. You give the players a choice with actual tradeoffs instead of making hazards uninteractive gotchas. However, when I say it's harder... it's a lot harder. A majority of scenarios are not written for time pressure to matter.
My players help with this one, because they typically feel a sense of urgency if the plot implies one. If people are kidnapped, they're already hesitant to spend a lot of time taking 10 minute breaks to heal. So naturally, they're hesitant to spend a lot of time combing the hallway for traps as well.
Now this I agree with, It is built with the idea that you arent forbidden to try again unless the action explicitly state so.
So we do know the developers intended a few things from how it is covered in both PC and GMC.
A. With enough time on their hands characters attempting non-consequential tasks automatically succeed and GM tells you how long it took. This is explicitly mentioned with search. As long as they can possibly do it, they will.. eventually. You could roll until you get a success.. or find the range between when the first success can possibly happen and when its functionally guaranteed to have happened. or any other random method of determining the time consumed by this.
B. GMs are encouraged to make the decision what can or cannot be retried. Seek and Search both being direct subjects of this reccomendation.
C. Hidden hazards, doors and objects in inconspicious spots typically cannot be found without meticilous and slow search.

NorrKnekten |
I think the disconnect here isn't about how the math works, but in appetite for success rate vs character investment.
If you use the Search activity you get only one chance of finding a trap before triggering it. Since most traps will be within a few levels of your own, the odds of success are somewhere in the 40% to 80%.
Deriven however feels that if you invested heavily in it, you should be almost certain to find the trap when looking for it. If you roll multiple times, it basically comes down to the chance that you'll roll a 1/critfail before you roll a success. Overall that gives far better odds than if you only rolled once.
I don't think this is how the game is intended. Traps are bad enough that if you're allowed to improve your odds this way, you should always do that, and the Search activity is so much worse that you should never use it.
Also, I don't think PF2 is designed with the idea that you can get so good at something that it'll almost always succeed. If you have to roll for something, there should be a decent chance of success and a decent chance of failure. By rolling multiple times you undermine that design.
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Also, if you can retry Seek indefinitely, then with one feat, Ageless Patience you basically "solved" all non-combat Seek problems, because you can never critically fail.
But.. seek doesn't have a critical failure part to it. Dont get me wrong I think Ageless Patience is intended to be frequently used for Seek but if we can just reattempt it either way then Ageless is a pointless waste of time. I absolutely love the feat though.
I could've missed something about critical failure imparting a false sense of security as written somewhere but I don't think I did.
Either way. I agree, This is not how the game is intended.

Deriven Firelion |

I want a perception based character to find traps and hazards nearly 100 percent of the time with rare exception, yet I still want a small chance of failure. Thus I allow multiple rolls. I would say the same of an Athletics or similar type of thing for something like breaking open a door.
The game is set at a roughly 50 percent failure rate, sometimes this rate is higher even for characters built for perception. Then there is variance which can lead to unlucky die rolls for a particular activity like trapfinding which makes the character look even worse than they are.
Recent example is a dungeon that had four traps. The traps DCs were set very high so you had a 50 percent or higher failure rate even for a character with a high perception, an item, and with trapfinding.
So you have this situation where the high Legendary Perception character with an item bonus will find maybe two of the traps. With variance if they start off missing the first trap, there are three traps left with a similar or worse chance of failure. The idea behind the percentage they should find 50 percent of the traps, but with variance they may find none of them because the probability each time is 50 percent or higher failure rate. With variance the d20 may not cooperate with that 50 percent probability.
Why? Because you're rolling hundreds of d20 with often four or more characters each rolling hundreds.
So the distribution of low rolls may occur when trapfinding leading to a character with an immensely high perception looking like Mr. Oblivious even when they have heavily built to detect things.
So NorrKnekten is telling me that is better narratively to roll once even if the Perception character misses all four traps looking within the narrative like a he can't find his nose on his own face just because the d20 distribution looks something like the following:
Mr. Perception: 2 20 17 12 7 1 4 18
You look at the distribution and it's supposed to give you a roughly 50 percent success rate but due to variance the following occurs:
Trapfinding Distribution: 1 7 2 4
Saving Throw Distribution: 20 12
Hit Roll Distribution: 17 18
Mr. Perception looks great swinging his blade.
His saves are really good or ok.
His trapfinding looks atrocious.
Thus I've found allowing multiple rolls to find traps works far better to smooth over any negatives caused by variance by allowing multiple rolls so Mr. Perception looks like Mr. Perception in the narrative rather than looking like Mr. I Can't Find a Trap Even Half the Time.
Narratively speaking it looks absolutely terrible when a character with Legendary Perception who has built up their wisdom with an item and feats like Trapfinding can't find the traps due to the generally 40 to 60 percent failure rate built into the game by allowing one roll with a bonus that doesn't account for a really low roll.
Multiple rolls flattens the variance and makes Mr. Perception look like Mr. Perception.
That doesn't even take into account even if they find the trap, they have to figure out how to disarm it or destroy it without setting it off and taking the effects. The DC of that is also usually set very high, but rerolls are built into skills like Lockpicking and Disable Device.
Not real sure why NorrKnekten can read and allow multiple checks on an Open Lock or a Disable Device which is clearly allowed in the rules, but somehow Searching needs be a single roll where if they fail multiple times makes Mr. Perception look narratively worse meaning if you were writing the story Mr. Perception would have walked his party into multiple death traps.
Can you imagine a character like a Sherlock Holmes or a Wolverine or a Daredevil failing all the time to detect the traps or hazards. It would make the PC look terrible when they have specifically built for detection.
That's why I moved to multiple rolls because the bonus still allowed too much variance. The multiple rolls led to a much higher success rate while still not making it 100 percent and led to the character being able to use the thievery they built up super high to disarm the trap rather than walking the party into half the traps in a dungeon and then possibly failing to disarm the other half.
I'm not sure what a narrative looks like with NorrKnekten's method. It feels like Mr. Perception in his games would look more like Mr. Maybe 50 Percent if the multiple D20s fall right so the low rolls aren't occurring on the trapfinding.
I'll leave it there for now. I know the method I use narratively rewards characters who build for Perception. If I want to make sure everyone can't roll, I set the Perception high enough (though this is often already done by the module designer) high enough only certain PCs even get a chance to roll. The flat bonus doesn't account well for variance given the sheer number of d20s rolled by an entire party.
It's why some of my players can have a real bad night when the d20s are falling badly for them on key rolls even though they have a 50 percent or better chance to hit. It hits harder for classes that rely on a single attack roll in a round in an all or nothing hit which is what trapfinding is in NorrKnekten's narrative. I don't think that makes the perception based characters look to great when the dice don't fall their way. I don't even think it makes them look good when they dice perform for the standard as finding the two death traps and having two hammer the party still makes Mr. Perception look terrible narratively.
I encourage anyone else running the game to reduce variance by allowing multiple rolls if the failure effect is non-existent or low so your players actually look good when they build something up to Legendary and invest heavily in it. It reduces variance and makes your players look competent at some skill or ability they've invested in. The PF2 DCs are often set too high making players look less than competent as no one in a story would enjoy watching some character built up as Mr. Perception or Mr. Strong failing to show their characteristic even 50 percent of the time.