People all the time are saying that +1 to hit is a 5% better chance of hitting. I assume this is because when you roll to hit you roll a d20 and 1 is 5% of 20, but that doesn't really make a whole lot of sense.
If you have a +33 attack bonus v.s. an AC 18 creature and then you get another +1, you have increased your chance of hitting the creature from 95% of the numbers on the die to 95% of the numbers on the die, which is a 0% increase. Similarly, if you have a +0 attack bonus v.s. an AC 30 creature an additional +1 provides no meaningful percentage increase in hit chance.
It's not just in extreme cases that this doesn't add up, though. Let's say that you hit a target on a 16+. If you get a +1, you now hit them on a 15+, which is a 25% increase in your chance of hitting, and a 6.25% reduction in your chance of missing.
It seems like the only time the first partial derivative of your chance to hit with respect to your attack bonus is 5% per point of difference is when you have just barely achieved a 100% chance to hit. In that situation, a 1-point decrease changes your chance of hitting from 100% to 95%, which is, in fact, a 5% decrease. That's an extremely unusual situation, given that it requires you to have some method of not critically missing when rolling a one, so it seems unlikely that that's a default situation in everyone's heads or something.
Nonetheless, I regularly am told that +1 on a d20 is a 5% increase in chance of success, especially in the context of attack rolls and AC. What do people mean when they say this, and is what they mean true?
Equivalent example for early edition players:
You are fighting an Orc with an AC of 7. You have a THAC0 of 15. This means you have a 40% chance to hit the Orc. If you had a +1 bonus to hit (for example from a magic weapon), you would have a THAC0 of 14 and a 45% chance of hitting the same orc, which would be 12.5% more likely to hit and 8.3̅3̅3% less likely to miss. How is this 5% better?