StuG G or StuH42?

There are a couple of StuG Battery army lists in Flames of War that allow you to choose between a StuG G or a StuH42 at no cost (usually only one per platoon). Looking over the two models you see that they have the same armor, same movement, same classification (fully-tracked, fully armored), same gun rate of fire, everything the same except three things. The G has an anti-tank rating of 11 with a firepower rating of 3+ while the H has a 10 and 2+, respectively, plus has smoke. So, which should you take?

Setting aside the smoke capability for the moment, as this blog entry is focusing on the math aspect of the choice. See, all of this came about listing to the Radio Free Battlefront podcast, episode #??, when one of the hosts mentioned that they like to use one particular over another because, even though it had a lower anti-tank rating than some other model, it had a 2+ firepower and when they hit it, they wanted to make sure it was dead. The implication was that the trade-off between anti-tank rating and firepower favored the firepower side. This reminded me of my wondering whether to buy the StuG or StuH for a new unit. (I bought the StuH so it would have some variety. I has since decided to buy and paint both, so I would always have a choice.)

So, which is better? The What Would Patton Do boys would tell you to do the math and figure it out, so I thought I would do just that. (Please note that I am not a mathematician by any means, but I think these numbers are "close enough".) So, lets start with the basics.

Versus Armor

First off there is the chance to hit. Given that both assault guns have the same range and rate of fire, any modifiers that apply to one would apply to the other. So, for simplicity, we are simply going to throw this variable out of the equation as they are equal, and thus cancel each other out.

Once a hit is obtained, the target rolls a 1D6 and adds the value to their armor rating, then compares the sum to the anti-tank rating (AT) of the firer's gun.

• If the sum exceeds the AT, the shot bounces and there is no further effect to consider. 
• If the sum equals the AT, the firer rolls 1D6 looking to equal or exceed their weapon's firepower rating (FP). 
• If the FP roll is made, the target is bailed out.
• If the FP roll fails, the shot bounces and there is no further effect to consider.
• If the sum is less than the AT, the firer rolls 1D6 looking to equal or exceed their weapon's FP.
• If the FP roll is made, the target is destroyed.
• If the FP roll fails, the target is bailed out.

So, to calculate the change to destroy a target, it is basically:

Chance to Exceed Target's Armor and Roll * Chance to Roll FP

The chance to bail out a target is calculated as:

(Chance to Exceed Target's Armor and Roll * Chance to Fail FP) + (Chance to Equal Target's Armor and Roll * Chance to Roll FP)

Using this information, the answer of which is better became clear: it depends.

Because the FP of the H is higher and represents the chance to "confirm" a penetration, the greater the chance of penetrating, the more likely the H would have a greater chance to destroy the target than the G. Put another way, if both the G and the H have a 100% chance of exceeding the target's armor and a 1D6 roll – say when firing at an M3 Stuart with its 3 armor* – then the H's 83% chance to roll the FP means it will out-perform the G with its 67% chance to roll the FP.

* Armor of 3 + the maximum roll of a 6 = 9. This is less than the AT of both the G and H, so both have 100% chance to penetrate.

As you work out the math against various armor ratings a pattern emerges. As the armor rating increases, the H's chance to destroy falls faster than G's does. In addition, the G always has a greater chance to bail out the enemy because of its greater chance to fail FP rolls. As the armor rating gets high enough, what kills the H is its failure to penetrate in the first place, negating the higher FP, which depends upon penetrating in the first place.

So, the conclusion? If you are fighting against armor rating 4 or less, the H is better. Against an armor rating of 5, 6, or 7 the G is marginally better. But against an armor rating of 8 or 9, the G is significantly better. (Against an armor rating of 10 the G has a small chance to bail the tank while the H cannot at all, but the chance is not significant.)

Versus Dug-In Infantry

Another tactical problem I often see where the main gun's performance is being considered is whether to use the main gun or the MG against dug-in infantry. Everyone says you always want good FP for digging out infantry, yet I frequently see people giving up the main gun for the MG when it comes to these situations. Why?

• 1D6 at a higher chance of total success can only produce a maximum of one lost Team, while rolling more dice at a lower chance of total success can potentially produce more lost Teams. Do you feel lucky?
• As with the armor example above, FP only comes into play if you hit first. So, the increase in FP has to sufficiently offset the increased chance to hit.

What makes this scenario harder to calculate is that you have to take in the chance to hit and the rate of fire (or movement) into account this time. What you can ignore is the chance for a failed save (3+ for infantry) as it is the same chance in all cases.

So, let's start by reviewing the process of hitting and killing dug-in infantry.

• Use the target's skill to find the basic chance to hit.
• Find the modifiers that apply.
• Grab the number of dice equal to your rate of fire (ROF).
• Roll the dice. Those that equal or are higher than the required number are hits.
• Infantry Teams get a chance to save against those hits. The chance is 67% (3+ on 1D6).
• As the infantry is dug-in, it counts as bulletproof cover, so the firer must confirm the unsaved hits with an FP roll.
• If the roll is less than the FP rating, the Infantry Team is unaffected by the unsaved hit.
• If the roll is equal to or greater than the FP rating, the Infantry Team is destroyed.

With the process defined, let's run some numbers. The biggest problem to solve is what the chance to hit is. The variable is not so much the target's skill or the modifiers, but the firing weapon's rate of fire, as the skill and modifiers are all equal for all weapon systems (with the exception of shooting at long range, but we will get to that). Thus the main variable to consider is the rate of fire of the weapon. Given that an MG's ROF is always 3, whether moving or stationary, but the main gun of a tank is not this will be a prime decider in whether to use MG or main gun. However, given that we are trying to determine whether the StuG or StuH is better overall, and the MG stats are exactly the same between the two, we need to remain focused on the main guns as much as possible.

So, how can you compare the different situations? If you think about what the chance of hitting with all three dice when you need a six it is basically 1/6 * 1/6 * 1/6 or 0.4%. So we tend to think of things as a percentage chance to hit. But, in this case it is a percentage chance for three hits, so how do we represent that? The easiest way I know is to simply call them fractional hits so you would multiply 0.4 * 3 for 0.013 hits. Not the best way, of course, but something I can understand. Suffice it to say it is a number and the higher the value the better.

So, looking at the spreadsheet (it has values for hitting on 2, 3, etc.) and basically the MG will never out-perform the StuH42 main gun when it can fire twice. The extra shot from the MG just does not offset the FP rating of the 10.5cm assault gun! Interestingly, the MG does the out-perform the 7.5cm gun if it gets both shots two. Where the difference comes in is if the main gun only gets one shot. Only then does the MG out-perform the main guns, and it does so for both.

The point of this exercise was to determine if maybe the MG out-performed only for the lower to-hit numbers, which would give more favor to the StuH over the StuG, but as it out-performed a moving main gun and under-performed a stationary main gun, no real advantage is given in moving fire. That said, as the AT rating does not come into play when fighting infantry, the StuH clearly out-performs the StuG when stationary, firing at dug-in infantry.

Whew!

Bring on the Smoke

Smoke adds another dimension that is really hard to quantify. Basically you can trade a shot to try and penetrate for one that will obscure a target. Smoking a Big Bad Target, to force them to move next turn is always a good option, so an advantage should go to the StuH. After all, options are better than none.

Summary

All in all, I like the StuH, but of course it depends upon what you expect. I think the trade-off is minimal, except against heavily armored targets, but then those might be the one you might want to smoke anyway!

Now, if Battlefront would only make the model...

Hey, let me know if there are other trade-offs out there like this; preferably one that is a 'no cost' replacement, or minimal in points. I think it might be fun to think it through. Also, if you see something wrong with the math (which I don't show...), let me know. I won't be offended. I would love – short of going back to school – to really learn how to calculate the odds. If only someone had an easy-to-use program... :^)