No problem. Thanks for commenting.

1) Yes, a success probability of 75% (e.g. DC16 in a d20 based system, after all adjustments have been added), means that for a large number of attempts 3/4 of them will succeed and 1/4 will fail.

While not directly relevant for a single test (at least no more than claiming something is “swingy”), over a course of a game that same DC16 test will come up a lot, e.g. hitting AC 16, a DC 16 test to jump, etc.

It is relevant mostly to TTRPG game design, such that it sets the tone of the system — if an averge character suceeds a lot it has more of a “heroes win” tone, but all the time might be boring. The failure chance is how often bad things will happen / characters fail / unexpected plot twists.

During a game it also informs characters whether the activity they are going to take is risky (hopefully high risk/high reward) or a safe bet. Yes, you might suceed at a high risk task (or fail at a low one), but if you want the character to be cautious you may chose the lower risk/reward activites (their wins won’t be as high, but they will fail/twist less).

The main point of the article was however, a counter to the common adage that a d20 system is too ‘swingy’ compared to a 3d6 system… which is not true is the statistical sense.

There are differeneces, e.g. they may be affected differently by modifiers (a static +1 has a bigger effect in the middle range of a 3d6, but less at the ends), but ‘swingy’ isn’t relevant.

2) They are the same; the system used to generate the binomial result make no difference.

A binomial system ‘stabilises’ (usually measured by standard deviation / variance; or expressed as a confidence interval) at the same rate, irrespective of the underlying mechanism of generating the result.

Effectively, whether you are rolling 3d6, 1d20, a pool of dice, drawing cards, or picking out of a hat, if you have a 75% chance to succeed (after all modifiers) you could replace the source of randomness with a computer generated if random() < 0.75 (where random() generates a random number between >= 0.0 and < 1.0). Then, 100 trials has 75 expected success with a standard deviation of 4.33; a 95% confidence interval for 100 trials with 75 success is 0.75 +/- 0.745 (i.e. 0.657 - 0.8245) 1000 trials has 750 expected with a standard deviation of 13.69 (10 times the size, but standard deviation only increased 3x); a 95% confidence interval for 1,000 trials with 750 successes would be 0.75 +/- 0.0258 (i.e. 0.7222 - 0.7758), much narrower. But, like the probability for a single throw, these are all in advance. It is possible to generate 100 (or 1,000) failures in a row -- just very unlikely.