Chris, you seem to have a better grasp of probabilities, but otherwise you're doing exactly the same thing. It's really bizarre.

Pat, I’ll say it again... when multiple people are misinterpreting you in exactly the same way which is more likely?

Chris, I'll say it again: When we covered this, I told you I considered that, and gave four reasons contrary. And I can think of a 5th one: On technical issues, the crowd's answer tends to be completely orthogonal to the correct answer. To press your case, you'll need to present a case, not just make an unsupported assertion.

And I’ll sum it up again +1 may increase the odds of success by 50%, but they’re also only reducing the odds of failure by 5.7%, which isn’t terribly meaningful at all.

The fact is that when people look at a bonus they aren’t looking at how much it improves your odds by, they’re looking at what the new odds are. They aren’t seeing “my odds of success have gone up by 50%” they’re seeing “my odds have gone from 10% to 15% and so that +1 might matter once in 20 rolls.*”

You’re stuck on arguing the equivalent of “you’ll halve your odds of heart disease if you stopped eating meat” as statistically significant, while everyone else sees “meat = 0.2% chance; no meat = 0.1% chance” and getting a cheeseburger.

No one is arguing your numbers are right; we’re arguing your numbers aren’t important to how people are evaluating success/failure and when most people say “my odds have gone up 5%” what they actually mean is “this distribution of possible results on this particular roll has a 5% greater chance of success and a 5% lower chance of failure than a distribution without the +1 bonus added to it.”

Only “my odds have gone up by 5%” is faster to say and most people know exactly what the speaker means by that and don’t feel compelled to correct them for it.

* or matter after only 5 rolls or not matter in 40 rolls because the flat probability of a d20 will only get close to the mean in many times 20 rolls... the odds of actually getting a perfect distribution of every one of the twenty results over the course of 20 rolls is astronomically small.

Now that appears to be a reason, but I doesn't seem to make much sense.

Let's start with your first one. If you're at the place on the scale where a +1 increases your chance of success by 50%, that's huge. If it's an attack, you're increasing your damage output by 50%. That it only reduces the chance of failure by a smaller amount is irrelevant. What is relevant is that, if your improved chance of success is only 3 in 20, then the game will be very swingy. With only a 15% chance of hitting, your damage output will vary a lot, with long stretches where you do no damage at all, and then a big jump when you hit. In contrast, someone with the same average damage who hits 85% of the time will be the slow and steady type, dealing out a smaller amount of damage fairly consistently over time.

In fact, that's one of the design strengths of old school D&D. You don't just get new powers as you level up, but the very nature of attacks and damage shifts, as you move from a highly swingy game to a more stable and consistent one. This creates a lot of variety in play, over the course of a PC's career.

You addressed the swinginess, but what does that have to do with looking at the probabilities? They're separate issues, and both are important for determining how a game plays out. It's like I tried to explain the Pythagorean Theorem, and someone objected that the walls are too thin to bear the load. That may be a valid issue, but the one doesn't contradict the other in any way. The frequency a bonus comes into play is the same: Valid, but not directly related to anything I said. The same is true regarding player perceptions, and how they relate to the game mechanics.

So why the objections with what I'm saying? I'm talking about a set of tools for correctly assessing changing chances of success or failure. I haven't made any grand claims that it's the only important factor. It's like there's some instinctive aversion to even allowing someone to talk about the matter.

It's also irrelevant that people make blatantly incorrect statements in casual conversation. I'm discussing a technical issue, and in that context, being precise is useful, and statements that are incorrect should be corrected. Again, it's perfectly valid to talk about how players perceive things, but it's not an objection to anything I said.