Saturday, February 19, 2011

Basic Probabilities - A Guide, Part II

This is a continuation of this post, where we cover the very basic steps required to determine the likelihood of a weapon wounding it's target.  Now let's take a look at something a little more in depth.

Let's say we are a Space Wolves player and have a few Long Fang squads in our army. We want to know what the odds of destroying a Rhino are when firing 4 Krak Missiles at it. A reasonable question.

Well, we can find out the odds of destroying a Rhino are from firing ONE krak missile, using the method we outlined above.

Step one: "Hitting Gate." As in the above example we are BS 4 so we hit on 3+ giving us a 2/3 chance of hitting. Good so far?

Step two: Now instead of a "Wounding Gate" we have a "Penetration/Glance" gate. This one is a little bit trickier but let's think it through. For vehicle damage you have to calculate the Glancing hits and the Penetrating hits separately and then add them together. In this example we will glance on a roll of a 3 (Strength 8 Missile plus 3 = 11), and penetrate on a roll of a 4, 5 or 6.

Let's start by looking at the penetrating hits and come back to the glances. We penetrate on a 4 - 6 so one half. 2/3 (hits) times 1/2 (pens) gives us 1/3 penetrating hits per shot fired. So far so good.

Step two (b): Cover Save Gate. Depending upon whether we want to say the vehicle is in cover or not, we can ignore this step or not. For this example I'm going to say that the Rhino IS in cover so that you know how to do it. So to get through the cover gate we need to have our opponent roll a 1, 2 or 3, so one half chance. Take our one third from Step two above, times one half gives us one sixth chance to get through all the gates so far.

Step three: Vehicle damage table gate. Now, the Krak missile is neither AP 1 nor AP -, and the Rhino is not open topped. So this is a straight roll on the damage chart, which is simple. Remember that in this example we want to know what the odds of destroying the Rhino are. To destroy the Rhino (ignoring the possibility of cumulative damage from immobilized/weapon destroyed results) we need to roll a 5 or a 6 on the damage chart. So one third chance of getting a destroyed result. Take the one sixth from Step 2(b) above times one third gives us one eighteenth, or about 5.56% chance per Krak missile fired to kill the Rhino.

And you guys wonder why Mech is King, lol.

But what about the glances? Well, in this example we don't care about glances because we cannot get a destroyed result with a glance in this case. But let's say we actually wanted to know what the total chance to get an "Immobilized or Better" result was, we would need to go through Steps 2, 2(b) and 3 to get the odds for the Glance, and then ADD that to the odds from the pen for the TOTAL chance per shot.

Pressing on.

So we have our probability for a Krak missile to kill a Rhino in cover: 5.56%. So what are the odds of FOUR missiles killing a Rhino in cover?

This is where a lot of people fall off the rails. Let's see if we can get them back on track.

Many people will tell you to take the 5.56% and multiply it by four to give you a total of 22.24% .

That is wrong, wrong, wrong.

Think about that for a moment. If we have four missiles and multiplied by four to get 22.24%, what would we get if we had 18 missiles fired? 100%? What about 20 missiles fired? 111.12%? That's absurd. We KNOW that even if we fire 18 missiles, we might still fail. Sure the odds are low but it's a possibility.

Let's talk about finding out EXACTLY what those odds are.

The way we find out what the odds of destroying the Rhino are is by finding out what the odds of NOT destroying the Rhino are. Don't worry it makes sense here in a minute.

So if the odds of destroying the Rhino are 5.56% with one shot. What are the odds of NOT destroying the Rhino? 94.44%? Right. Now we just have to multiply the odds of it not happening once by the odds it not happening twice times the odds of it not happening thrice times the odds of it not happening four times.

So .9444 times .9444 times .9444 times .9444. Another way to think of it is .9444 to the fourth power.
What do we get when we do this calculation?

We get .7955 (or 79.55%).

That's the total odds that we will not wreck the Rhino if we fire at it four times.

Well if we know the odds that we WON'T wreck the Rhino, don't we also know the odds that we will?

Of course we do. It's 100% minus 79.55% equals 20.45%.

So if we fire ONE missile the odds of wrecking a Rhino in cover are 5.56%

If we fire FOUR missiles the odds of wrecking a Rhino in cover are 20.45%

Quite a bit higher, yes? But not as high as the "5.56 times four" method, which would have given us something higher than reality.

Notice that no matter how many shots you fire (how many times you multiply .9444 times itself) the odds of you "not destroying" the Rhino will never reach zero. The number will keep getting smaller and smaller but never reach nothing. Likewise, the odds of you destroying the Rhino will get larger and larger but will NEVER reach 100%. And every missile shot that you add contributes a little less to bringing up your chances than the missile before.

It's called "diminishing returns" kids. ;D

In the next part we will add another wrinkle to the puzzle: meltaguns and multiple meltaguns!