Is probability of succeeding an opposed check the same when the second check is passive?

Question

An "opposed check" is a well-known kind of check in many games, when two opponents roll dice, add modifier and compare results. It differs from a convenient dice roll when you compare your result against a constant number (DC, target number or whatever).

There is also a concept of "passive check" in 5e when you just take 10 instead of rolling d20. An opposed check against a "passive" value effectively turns into a "simple" one, being made against a constant number.

Now, let's say there are two sides makes opposed rolls using modifiers A and B. First one rolls d20 and adds A, while second one always uses constant number 10 + B. In terms of statistics, should these checks have the same hit/miss rate? —

1. d20 + A as opposed to d20 + B
2. d20 + A as opposed to 10 + B

I'd say rolling d20 is a little better than 10, since the average of d20 is 10.5, but my probability intuition isn't very good.

In order to minimize number of dice rolls, if I change all opposed PCs vs. NPCs checks to "roll vs. passive" checks (players always roll, NPCs always use their passive values), how does it affect my games?

Answer 1

Let's first simplify the problem by setting X = A − B. It should be clear that the success rate of d20 + X vs. a plain d20 is the same as for d20 + A vs. d20 + B. Similarly, the success rate of d20 + X vs. 10 is the same as for d20 + A vs. 10 + B.

It turns out that, for X close to 0, both of these methods give similar odds, with the roll vs. 10 being slightly more likely to succeed due to the fact that, as you note, the average of d20 is 10.5 rather than 10.

However, for sufficiently large X, rolling against 10 is a lot more likely to succeed. In particular, X ≥ 9 or higher obviously always succeeds against 10, but the success rate of d20 + 9 vs. d20 is only 86.25%.

Conversely, for sufficiently negative values of X it's better to roll against a d20 rather than against 10. In particular, X ≤ −11 always fails against 10, but d20 − 11 vs. d20 still has an 11.25% chance of success.

Looking at the output of this AnyDice program, the break-even point turns out to be at X = −5, where both methods give a success rate of exactly 30%:

So, in conclusion, rolling d10 + A vs. d20 + B and rolling it vs. 10 + B are approximately equivalent as long as the difference between the bonuses A and B isn't too large. But if the difference gets close to ±10 or more the two methods start to give significantly different success rates.

Answer 2

It makes things easier for the side rolling dice

Unless the net difference between A and B is equal to -5 when it’s the same or less than -5 when it’s worse. How much better varies with the modifier. Here is an anydice program you can play with.

Your intuition that because one has a higher average (both median and mean) was pretty good but that’s only part of it. The fact is the passive check gives a linear result but two dice are non-linear.