Hit Chances in d10 SAS

It's a sad fact, but probability does come into play in any Silver Age Sentinels game -- unless you go diceless -- but how those probabilities shake out is not always obvious.

In another thread, I made this statement:

Keep in mind that an 11 ACV is a 55% chance to get a chance to hit. Pair it against a 11 DCV, and you're looking at actually making your hit (i.e., you make your attack roll and they miss their defense) about 25% of the time.

Well, that sort of thing gets one to thinking, and so, I wrote a quick little thang that spit out some numbers for me comparing ACV's of 2-19 to DCV's of 2-19 and calculating the percentage chance to hit (defined as a successful ACV check and a failed DCV check).

You can view that matrix now at: http://www.evilhat.com/sas/hitmatrix.pdf

I've color coded the cells in the table along the diagonal for a particular reason -- that diagonal is the path along which a Trick Shot moves -- in that for each -1 your ACV takes, the opponent takes a -1 to their DCV.

The results can be pretty interesting -- and they go to show that where your 'hump' is in terms of what's the best trick shot penalty to take when stacking a given ACV against a given DCV.

Now, if you look along the other diagonal, from the lower left corner of the matrix to the upper right, you'll see that the 'hump' (best probability) lies along the line you can draw between those two corners.

So, there's some pretty simple math you can derive to determine what your best hit chance is going to be -- it's the closer the sum of your ACV and your opponent's DCV gets to 21.

So, if I am playing a guy with a 15 ACV, who's going after someone with a 10 DCV, that puts me at a total of 25 -- 4 above 21. To get the best chance of hitting that guy, I should take a trick shot penalty of -2 (half of 4), since that drops my ACV to 13 and his DCV to 8. Looking at the matrix, sure enough, that improves my chances by about 5% (from 46% to 51%), which is as good as it can get. If I take any more of a trick shot penalty, I'm hurting my chances.

Now, originally, I'd thought that the simple Trick Shot rule of thumb was 'don't drop your own ACV below an 11', since an 11 represents a 55% chance of success (without factoring in DCV chances). This matrix disproves that.

Here's another interesting realization. The hump tends to fall in an interesting sort of one-bend line for a given ACV, as you 'flex' the DCV value. We don't have a rule for /increasing/ your ACV in order to increase your opponent's DCV

For illustration, let's go back to that ACV of 15.

vs DCV's 2-7, the hump is right where I'm at; trick shots won't help, they'll hurt.

vs 8, a -1 will help

vs 9, a -1 or -2

vs DCV 10, a -2

vs DCV 11, a -2 or -3

vs DCV 12, a -3

vs DCV 13, a -3 or -4

vs DCV 14, a -4

and so on.

Okay, so this shows us that you can determine what your sweet spot is for one ACV and derive a rule of thumb from there.

For my ACV 15 guy, my rule of thumb is '-1 against a DCV of 8'. I know that I can (and should) take an additional -1 for every 2 points of DCV my opponent has over 8. Want to get my best chance at hitting someone with a DCV of 20? That's 12 beyond 8, so I should take a -7. Sure, I'll be rolling against an 8, but he'll be rolling against a 13, and checking the matrix, 8 vs 13 is 7.8%, the best chance along that particular diagonal.

Yeah, hitting guys with that kind of DCV sucks a bunch.

I'm sure other principles can be derived as well from observing the data on the chart -- there are a number of cases, in lower ACVs, where taking trick shots just won't help you, ever. ACV 9 is a good example for this. Since you can't move down-and-right on a diagonal, only up-and-left, the 'hump' for our ACV 9 fella is always out of reach until the opponent's DCV goes over 12 (9+12=21, so we're back to that original principle).

So, in summary:

• Never consider taking a trick shot if the sum of your ACV and your opponent's DCV is lower than 21.
• If your opponent's DCV + your ACV is greater than or equal to 22, the penalty you should take to get your best chances of hitting on a trick shot operates on the following formula:

( Your ACV + Their DCV - 21 ) / 2

So if I've got an 11 ACV, and I'm facing down someone with a 13 DCV:

( 11 + 13 - 21 ) / 2 = 1.5

As luck would have it, it doesn't matter which way you round -- a -1 or a -2 will get you the same percentage chance of hitting -- and under these circumstances, that's hitting 16.2% of the time.

All other things being equal, when facing a given ACV-DCV combo, there's only so good you can get.

Some related observations, in closing:

• You are, indeed, meant to "whiff" a lot in a fight, from looking at this matrix, if your opponent can defend. Chewing away at his Defenses, so those -4 penalties stack up, is incredibly key.
• If you have a complaint about the frequency of whiffing in the game, tone down the Extra Defenses -- and if there are no Extra Defenses, you may want to consider a 'nobody gets a free Defense' rule, which starts everyone off at a -4 right out the door.
• If you have a problem with the genie I've let out of the bottle by writing up this bit of math, quash it by keeping your NPC's DCV's secret! Even if someone's got the Judge Opponent combat technique, you shouldn't give out the hard numbers if you can help it. Make them sweat a little. (And as someone pointed out in response to this, ALWAYS make your players describe the trick shot -- if it's not tricky, it's not a Trick Shot!)