If it were following a bell curve, then the yellow circle would have ~66% of the bullets landing in it not 50%.

For fifty percent you need a flat distribution. The radius of the yellow circle is half that of the red circle, which means the area inside the yellow circle and outside are the same. That means that for both areas to have 50% odds of hitting, the distribution has to be flat.

If the distribution was a truncated bell curve, then the circles correspond to the standard deviations, and that’d mean that ~66% of the bullets would appear within the first standard deviation.

Why would it equal 66%?
Wouldn’t the highest middle point on the bell curve be the edge of the yellow circle? Where the curve going up being equal to the curve going down is representative of the chance for bullets in the centre yellow circle being equal to the chance for bullets in the outer red circle?

Like in that picture I linked. There’s 34% chance on either side of the middle point of the curve, then down to 13.5% and 2.5%. Meaning that on both sides there’s the same percentage chance.

I said approximately 66% as I was recalling from memory. It is more accurately 68.2%.

The yellow circle would have both 34.1% chances because the exact center of the bell curve would be the exact center of the circle. The left side of the circle, would get the left 34.1%, and the right side would get the right 34.1%

This is not how it works. As explained in this post by UnstableVoltage: Shots too random.

“The spread within the cone works on a bell-curve with highly inaccurate shots being as unlike as highly accurate shots (meaning that most rounds will tend to fall somewhere close to halfway between the red circle and the bullseye).”

In other words the centre of the yellow circle is the edge of the bell curve since it’s just as unlikely as the outermost point of the red circle, since these are the “highly accurate” and “highly inaccurate” shots he’s talking about.

So, really it is 50% for each circle but it’ll be a tight spread around the ring of the yellow circle with very few shots landing in the centre of the yellow circle or on the extremities of the red circle.

I believe there must be some confusion here; let me try to clear things up a little:

The accuracy of a shot must be measured by its distance from the center of the circle(s), i.e., its radius, and the amount of shots of a certain accuracy is then the integral over the circumference of circles of that radius – an integral whose terms increase linearly with this radius.

This means that if shots are uniformly distributed within the red circle – what I must assume is meant with this flat distribution – then each shot has a 25% chance to land inside the yellow circle, since it has a quarter of the area of the red circle. It also means that extremely accurate shots are much less likely than extremely inaccurate shots, since the radius of accurate shots is much smaller than that of inaccurate ones.

It also means that if the distribution is bell shaped, with its maximum in the centre of the circles, it’s still quite possible both that 50% of the shots fall inside the yellow circle – in fact the function has to have a higher mean value inside of the yellow circle than outside it for this to be true – and also that most shots will fall close to the yellow circle – even though the highest density of the distribution is at the centre! Remember, the radius increases outwards, while the probability density decreases (if it’s a centred bell curve), and the most common accuracy will be where the product of these two functions has its maximum; intuition is that this may coincide nicely with the half-radius, yellow circle.

After all, this is such an intuitive way to implement inaccuracy I’d be highly surprised by any other choice

I feel this is way too complex and hard to calculate with. I understand the game tries to be different from the current XCOM games, but let’s remember, all the old X-COM games too had percentage hit chance displays when you aimed, even though the bullets/plasma bolts followed the laws of physics and could be impacted into objects between you and the target. I feel while the aiming reticle is a nice touch, it should just be used for selecting target bodyparts, and the to hit chances should be displayed numerically.

Indeed, truncating a bell curve such that the inner half-radius disk gets 50% of the shots, means the most likely accuracy is at 0.43 of the radius of the outer circle, i.e., almost that of the yellow circle.

I guess they may also have decided that the probability as a function of accuracy / radius should be highest exactly at the half-radius, yellow circle, and compensated for the difference in radii inwards and outwards from there – this could, depending on truncation of the bell, give a probability density which was either a local minimum, maximum, or neither, at the centre of the circles. The resulting probability distribution could therefore be almost the same as the centred, bell-shaped one.

The position of each shot is perhaps most easily computed this way, as well, as it is simply choosing an accuracy according to a bell-shaped distribution, and uniformly picingk an angle at which to go outwards from the centre.

This was one of the things that I was just coming to post about! While I like that the enemy slightly moves around while aiming at them, would it be possible to have the game freeze their position once we hit the fire button? I had this exact same issue where the Queen moved her leg during the shot animation, causing a sniper round to completely miss even though the red circle completely covered it.

Even if the soldier were to keep aiming at the same point on the enemy when they shift, the shift alone can cause all sorts of issues depending on the angle of the shot. For example shooting at the joint of a back leg on the Queen, only to have that joint shift behind another body part or cover AFTER you have clicked fire could be the difference in your soldier living/dying. At which point I don’t think people would find it overly fun to lose soldiers (or missions!) because even though they did it right the game RNG shifted the enemy after selecting fire.

@JMPicard, You’re assuming a lot there. The developer that commented simply said that it was defined by a bell curve and that extremely accurate shots were just as unlikely as extremely inaccurate shots.

That second part is important because it tells us that the edges of the bell curve, the least likely results, are at the centre of the yellow circle and the extremities of the red circle. This means that the highest point of the bell curve, is the rim of the yellow circle, since there’s 50% in each circle.

While we’d all like to believe it’s a bit more complicated than that, it doesn’t seem to be at the moment. The reason I started the other thread on shooting patterns is because it didn’t feel accurate enough. In fact, when describing my issue I pretty much described my shots going around my target rather than hitting them. Which would make sense if the shots were concentrated just inside and outside of the yellow circle’s rim.

It’ll be obvious to you once you realise the bell curve you’re thinking about is the integral over concentric circles, whose circumference grows linearly with their radius …

@JMPicard
Between this and your other post it’s quite clear that you’ve got a very good understanding of mathematics, which I clearly lack, hahaha. I don’t know if you’re studying it or whatever but I really only have a basic understanding of maths. I don’t think I really understand the relationship between the theoretical calculus and it’s practical application, which is why I have no idea what you’re saying since you’re talking pretty much exclusively calculus.

What I do understand though is that this- “if the distribution is bell shaped, with its maximum in the centre of the circles, it’s still quite possible both that 50% of the shots fall inside the yellow circle” - doesn’t make sense.
If the centre of the bell curve is the centre of the circle then there would be a 66% chance for the bullets to land within the yellow circle, with the highest possible likelihood being the exact centre of the circle.

(either that or you’ve cut it in half down the middle and placed it end to end, where the centre has 33% chance and the outermost extremities have 33% chance. This also wouldn’t make any sense though. Why would the very outer edge be as likely to have bullets land in it as the centre of the circle?)

This would make the developer’s comment was wrong: “The spread within the cone works on a bell-curve with highly inaccurate shots being as unlike as highly accurate shots (meaning that most rounds will tend to fall somewhere close to halfway between the red circle and the bullseye)”

You also say at the end “After all, this is such an intuitive way to implement inaccuracy I’d be highly surprised by any other choice”, which suggests that you’re assuming this is the way they did it since it seems to be the most sensible way to do it in your eyes. However this doesn’t necessarily mean that this is the way they did it. From the developer’s quote it seems very likely that they actually do it a less sensible way, where bullets very rarely hit the targeted enemy or body part and instead are far more likely to hit around it.

Nobody “officially” said that the “center (yellow line) has 33% chance”. There are numerous “bell-curves”, and mapping that to a target-circle makes it interesting.

You have a small circle and a big circle. You can compare their radius and their area. If the [smaller circle] has a smaller overall area than the [big circle minus small circle] (so the “band” between outer and inner circle) but each bullet still has 50% chance of hitting the inner area, you have a (relative) “accurate” weapon, while if the area of both is the same, you have a “less accurate”.

We can compare the radius of both circles, but while that will tell us the same, it’s harder to imagine for most ppl, because we’re comparing a circle’s area to a “band’s”, which is hard to visualize accurately in your head.

The chance of hitting inside the circle is the sum (integral) of the area (between the curve and the 0 coordinate…line? I’m not native to english math…) of “the” bell-curve from one side to the middle, if the middle is the inner circle. There was nothing saying that you will hit dead-center the most with an accurate weapon, but that 50% of the shots will be inside the inner circle.

Saying that aiming with the inner-circle as “accurate” is a good idea is misleading. If you imagine the inner circle’s area as lots of circles, and every circle has x% chance to get hit, you must calculate with the length of each circle (circumference) too.

Which is better for you, standing in 1m^2 with 50% chance of a bullet hitting somewhere in it or standing in 100m^2 with 100% chance of a bullet hitting somewhere in it?

So I’m just saying that working with a bell-curve is not necessarily bad, if it’s done right. And we should have faith in the devs, we backed them for a reason.

Well I mean one can have faith in the devs but that doesn’t mean that one has to agree/like everything they do. I’ve somewhat covered this in my feedback thread about the system, but I’ve used a similar system in another game (something like 20% of all shots were guaranteed to hit the outer most circle) and it wasn’t very enjoyable. The bell curve is also counter intuitive for aiming but I’ll cover that later when I’m not on my phone and can get some in game SSes to illustrate my point.

Ok, yeah, sorry. There was no reason to think it would be 33% or anything like that. I think I was using it as an example earlier and got it muddled up with fact later, haha.

I understand what you’re saying about the area of each circle and how that translates to more or less chance when they have 50% each. When it comes to visualising the graph and the area under the wave as a bullet spread pattern it can be a bit hard to visualise. The way I understand it is that the first half of the graph is aligned with the inner circle’s (the yellow circle’s) radius and the second half of the graph is aligned with the red circles radius. So that the crest of the wave falls on the point where the inner circle and outer circle meet. Conversely the trough of the wave would therefore fall at the centre of the inner circle and the outskirts of the outer circle. The area under the wave at any given point would be the percentage chance for the bullet to be shot in that direction. This would all be divided by 360^ to give the bullet that radial spread instead of it being on a single axis.

Hmm, my issue isn’t the distribution of the shots. I’m fine with 50% being in both the yellow and the red, however at the moment, with the way I understand a bell curve and the way it would be integrated, it seems like the majority of shots will land on the meeting point between the yellow and red circles. This might not be so bad if, as you say, it’s only taking the very top portion of the bell curve so that the percentages are much more spread out rather than going from 0% to 50% it goes from 20% to 50% or something.

We don’t know exactly “which” bell-curve they use. It may be possible that the difference between Inner Circle and halfway between inner circle-bullseye is 1%. It may be 50%. We don’t know. And that’s my problem. people tend to imagine the “worst possible”, not one which may provide a fun game after balancing/tuning.

Isn’t it possible to have animations freeze in a default position when aiming and taking shots? Animations are good aesthetically, but having to wait until the animation has the enemy the most exposed before taking the shot really drags the pacing of the game.