Radical Alternatives: How SHOULD Size and Speed/Range Affect Chance to Hit?

[Rupert]

Originally Posted by Tomsdad

Yep, or even turning into them, if you haven't already done this to make yourself a herder target.

Turning into Torpedo spreads as well would I think count for this (as in trying to adjust your position so you are not where the attack is)

Turning into or away from torpedo spreads is more about hoping they miss either side, but can also mess up their target solution if you spot them very far away (not the common event). Usually you wanted to turn away, because most torpedoes weren't that much faster than a warship, so you might well manage to increase the range to the point where they ran out of fuel. All this assumes you weren't in a nice tight formation with friends, of course - in that case you just got to suck it up and take the hit.

[Ulzgoroth]

Originally Posted by Anthony

Naval gunfire. Flight time could exceed a minute in battleship duels.

But were those battleships capable of detecting incoming shells and predicting their point of arrival? In WWII, the last time battleships shooting at each other was a thing that happened?

I believe there was radar that could be used to direct guns and to spot the fall of your own shells at night. I've never heard of radar of that time being used to track incoming shells.

[Tomsdad]

Originally Posted by Rupert

Turning into or away from torpedo spreads is more about hoping they miss either side, but can also mess up their target solution if you spot them very far away (not the common event). Usually you wanted to turn away, because most torpedoes weren't that much faster than a warship, so you might well manage to increase the range to the point where they ran out of fuel. All this assumes you weren't in a nice tight formation with friends, of course - in that case you just got to suck it up and take the hit.

I'd argue that at that scale that's what dodging looks like :-)

(although it might not be just the one active defence roll in a vehicular combat system designed to show the detail of all that!)

[Balor Patch]

Originally Posted by Rupert

Turning into or away from torpedo spreads is more about hoping they miss either side, but can also mess up their target solution if you spot them very far away (not the common event).

The captain of HMS Repulse described dodging torpedoes as being "like an interesting game" until the Japanese finally got a good cross drop.

[Rupert]

Originally Posted by Tomsdad

I'd argue that at that scale that's what dodging looks like :-)

That's probably reasonable for many campaigns.

[Rupert]

Originally Posted by Balor Patch

The captain of HMS Repulse described dodging torpedoes as being "like an interesting game" until the Japanese finally got a good cross drop.

Aerial torpedoes, especially early in the war, had to be dropped quite close, and weren't terribly fast, thus you always knew where they'd been dropped (assuming good visibility), and the short range was countered by the slow speeds. Aerial torpedoes alone were quite poor at sinking battleships - it generally took a great many, plus lots of bombs (from dive bombers by preference) to sink a battleship that was free to manoeuvre. The real killer only turned up late in the war - remote controlled glide bombs, as used to sink the battleship Roma.

[Humabout]

Vicky, you make the assumption that shots fired approximate a uniform distribution, which is not the case. They will be clustered more densely around the actual target and become less dense as you look further out. So the size of the target won't have a quadraric effect on hit chance.

Let's assume a normal distribution of distance from bullseye with a standard deviation of 1 unit. If the target is 1 unit radius, the shooter has about a 68% chance of hitting. If we double the target's radius, he now has a 95% chance of hit. Adding another 1 unit to radius increases his odds of hit to 99.8%. He'll never hit 100%.

Now that addresses target size. Distance effects I'm not sure of. It would definitely increase the standard deviation of the final spread, but at what rate, I don't know. The shooter's own ability to be on target plus the effects of atmospheric conditions would be factors, though.

[Anthony]

Originally Posted by Humabout

Vicky, you make the assumption that shots fired approximate a uniform distribution, which is not the case. They will be clustered more densely around the actual target and become less dense as you look further out. So the size of the target won't have a quadraric effect on hit chance.

It's generally going to be a chi-square distribution, but that comes very close to quadratic once hit probability is low.

[Agemegos]

Originally Posted by Anthony

It's generally going to be a chi-square distribution, but that comes very close to quadratic once hit probability is low.

It's a bivariate normal, so the chi-squared distribution simplifies to an exponential distribution. In the general case the pattern is ellipsoidal, but assuming that the vertical and horizontal dispersions are uncorrelated and have the same variance you get the result that the chance id a given hit falling more than a distance from the centre of the pattern falls off exponentially with the distance.

[Eukie]

Originally Posted by Agemegos

It's a bivariate normal, so the chi-squared distribution simplifies to an exponential distribution. In the general case the pattern is ellipsoidal, but assuming that the vertical and horizontal dispersions are uncorrelated and have the same variance you get the result that the chance id a given hit falling more than a distance from the centre of the pattern falls off exponentially with the distance.

To be explicit:

P(Hit) = 1 - exp((2σ/x)^-2)

Where x is a measure of the target's projected area (typically this model assumes that the target is circular) and σ is the standard deviation of the projectile at the target ('projectile' should be interpreted liberally, and includes light beams and even the view through a camera).

Often we'll want to decompose σ into two parts: the angular dispersion α and the range r, or in other words:

P(Hit) = 1 - exp((2αr/x)^-2)

Multiplication is clunky, so we may want to define:

A = ln(αr/x) = ln(α) +ln(r) + ln(1/x)
P(Hit) = 1 - exp((2*exp(A))^-2)

P(Hit) is now a function of only one variable (A), which is a simple sum. This is broadly what GURPS already models: a ranged attack's skill, Acc, and other modifiers is ln(α), the Size Speed/Range table gives ln(r), and SM gives ln(1/x). Conveniently, 3d6-roll-under is a rough but workable approximation of 1 - exp((2*exp(A))^-2).

Where things get tricky is when we want to be more accurate about σ.