Alternate probabilty models
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 Quizatzhaderac
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Alternate probabilty models
When it comes to the probabilities of certain common random events, many people have beliefs that are simply wrong. However, it occurred to me that if a random event was being simulated on a computer, probability actually could work that way. For instance on a real die the probability of rolling a six is independent of the previous roll, but a virtual die could indeed have a memory.
So I was wondering what are some commonly believed probability mechanisms? How could they be built from classical mechanisms? Where might it be fun to actually have such mechanisms in games? Here's a couple I could think up:
Depletable dice: High throws decrease the future probability of high throws throws and increase the probability of low throws. Visa versa with low throw results. I can see this implemented by having each die have a set of outcomes like {1,1,2,2,3,3,4,4,5,5,6,6}. Which each roll one outcome is classically selected. When a number is "rolled", it is removed from the set and it's opposite is added; so after rolling a 5 the die's set would be {1,1,2,2,2,3,3,4,4,5,6,6}
Hot/cold streaks: Past results are an indication of future performance. So if you just got three heads in a row on a coin toss then a heads is probable on the next flip. This could be implemented by using the rule of succession on the past eight results. (minimum 10 % chance of a result)
Lucky agents: Events are weighted according to the luck of the agents that benefit from them. So if you have two agents A and B, with luck values of 2 and 1 respectively, an otherwise fair coin toss would favor A by 2 to 1. For events without a second agent, an agent's luck would be used against the average luck of all agents.
So I was wondering what are some commonly believed probability mechanisms? How could they be built from classical mechanisms? Where might it be fun to actually have such mechanisms in games? Here's a couple I could think up:
Depletable dice: High throws decrease the future probability of high throws throws and increase the probability of low throws. Visa versa with low throw results. I can see this implemented by having each die have a set of outcomes like {1,1,2,2,3,3,4,4,5,5,6,6}. Which each roll one outcome is classically selected. When a number is "rolled", it is removed from the set and it's opposite is added; so after rolling a 5 the die's set would be {1,1,2,2,2,3,3,4,4,5,6,6}
Hot/cold streaks: Past results are an indication of future performance. So if you just got three heads in a row on a coin toss then a heads is probable on the next flip. This could be implemented by using the rule of succession on the past eight results. (minimum 10 % chance of a result)
Lucky agents: Events are weighted according to the luck of the agents that benefit from them. So if you have two agents A and B, with luck values of 2 and 1 respectively, an otherwise fair coin toss would favor A by 2 to 1. For events without a second agent, an agent's luck would be used against the average luck of all agents.
The thing about recursion problems is that they tend to contain other recursion problems.
Re: Alternate probabilty models
Believable Coin Flipping: Alternating results seem more random than true random results. Give a 75% chance that the next flip will be different from the previous flip. (75% seems really high, doesn't it! When I tested it in excel, it seems very reasonable though, while 67% seems a bit too streaky!)
Emotion and Momentum: The concept that game scores are dependent on the emotional state of the participants. There would be a behindthescenes emotional rating for everyone involved, related to each participant's past performance. I'll make up some BS: At the very low end of performance, players would be angry and do better than expected. At the middlelow end of performance, players would be despondent and do worse. At the middlehigh end of performance, players would be excited and do better. At the very high end of performance, players would be cocky and do worse. So it would be a cubic function I guess, something like: Emotion Factor = X^3 + 2X where X is past/recent performance. The next Performance of each participant would be affected by their Emotion Factor.
I spent a while simulating that last one in excel and it looks like random chance with long stretches of hot and cold streaks.
Fate: If there's ever a situation where one unlikely result will make up for a stretch of bad luck, then that fateful result is much more likely.
Emotion and Momentum: The concept that game scores are dependent on the emotional state of the participants. There would be a behindthescenes emotional rating for everyone involved, related to each participant's past performance. I'll make up some BS: At the very low end of performance, players would be angry and do better than expected. At the middlelow end of performance, players would be despondent and do worse. At the middlehigh end of performance, players would be excited and do better. At the very high end of performance, players would be cocky and do worse. So it would be a cubic function I guess, something like: Emotion Factor = X^3 + 2X where X is past/recent performance. The next Performance of each participant would be affected by their Emotion Factor.
I spent a while simulating that last one in excel and it looks like random chance with long stretches of hot and cold streaks.
Fate: If there's ever a situation where one unlikely result will make up for a stretch of bad luck, then that fateful result is much more likely.
Adam
Re: Alternate probabilty models
Adam H wrote:Emotion and Momentum: The concept that game scores are dependent on the emotional state of the participants. There would be a behindthescenes emotional rating for everyone involved, related to each participant's past performance. I'll make up some BS: At the very low end of performance, players would be angry and do better than expected. At the middlelow end of performance, players would be despondent and do worse. At the middlehigh end of performance, players would be excited and do better. At the very high end of performance, players would be cocky and do worse. So it would be a cubic function I guess, something like: Emotion Factor = X^3 + 2X where X is past/recent performance. The next Performance of each participant would be affected by their Emotion Factor.
I played a game that was sort of like this once and really liked it. It wasn't that detailed, but for all confrontations, the two things rolled against each other and then had various modifiers added to it, and the difference determined the outcome. A pretty important modifier was what had just happened  if two entities are fighting and initially roll a 6 and 2, then the higher entity probably just killed the other. If it's a 6 and a 5, then it grazed it and the second entity suffers 1 on the next roll. If it's like, 63, then the entity suffers a penalty for the rest of the rolls in the chapter.
It worked really well, because a terrible roll or two would screw you up really badly, and starting to screw up makes being defensive a good decision.
 Will
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Re: Alternate probabilty models
I'm really interested in how these kinds of processes can be used in gameplay systems, and how that affects the players engagement. Think of Adam's "Fate" example in a typical D&D game  allowing the players to get their asses kicked in an important fight using a string of bad luck and then following it up with a lucky hit that wins them the fight, for example, has a strong effect on dramatic tension and excitement.
These two are actually at odds. Essentially, what you're talking about with depletable dice (and Adam's "believable coin flip" technique is the same principle) is making a random process conform to the Gambler's Fallacy. This will obviously make streaks less common in general, since the probability of some event A is reduced when the previous random process chose A.
Quizatzhaderac wrote:Depletable dice: High throws decrease the future probability of high throws throws and increase the probability of low throws. Visa versa with low throw results. I can see this implemented by having each die have a set of outcomes like {1,1,2,2,3,3,4,4,5,5,6,6}. Which each roll one outcome is classically selected. When a number is "rolled", it is removed from the set and it's opposite is added; so after rolling a 5 the die's set would be {1,1,2,2,2,3,3,4,4,5,6,6}
Hot/cold streaks: Past results are an indication of future performance. So if you just got three heads in a row on a coin toss then a heads is probable on the next flip. This could be implemented by using the rule of succession on the past eight results. (minimum 10 % chance of a result)
These two are actually at odds. Essentially, what you're talking about with depletable dice (and Adam's "believable coin flip" technique is the same principle) is making a random process conform to the Gambler's Fallacy. This will obviously make streaks less common in general, since the probability of some event A is reduced when the previous random process chose A.
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Re: Alternate probabilty models
Yeah, as far as streaks go, I'm not sure there's any way to model what the typical moron thinks. Is the less common result "due", or will the streak continue because of "momentum"? Hard to say. Maybe different participants should just have different tendencies.
In Warcraft 3 (and probably other games), critical strikes (like 20% to do extra damage) were modeled so that if a hero hadn't gotten a crit in a while, the next hit would be more likely to crit. I remember in DOTA some people thought it would be a good idea to hit creeps a few times to "charge up their crit" so that they would be more likely to crit against the next hero they fought.
In Warcraft 3 (and probably other games), critical strikes (like 20% to do extra damage) were modeled so that if a hero hadn't gotten a crit in a while, the next hit would be more likely to crit. I remember in DOTA some people thought it would be a good idea to hit creeps a few times to "charge up their crit" so that they would be more likely to crit against the next hero they fought.
Adam
Re: Alternate probabilty models
Combining streaks and depletion (or at least a certain amount of memory) could be achieved, though, by having a pool of streaks to choose from rather than individual rolls. So once you've used up the {1, 2, 3, 1, 1, 4} streak, you then might hit the {5, 6, 6, 6, 4, 6} one. Or maybe the {1, 6, 1, 6, 1, 6} one, for swingy luck.
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Re: Alternate probabilty models
City of heroes had (have?) a feature where you were guaranteed to hit after enough misses, depending on your chance to hit that foe. Iirc the statistical impact was tiny, but perhaps a bit less cursing the devs.
Re: Alternate probabilty models
ConMan wrote:Or maybe the {1, 6, 1, 6, 1, 6} one, for swingy luck.
I eat a sandwich! I swing at the orc! I spit on the ground! I climb the mountain! I look at Dwayne! I throw the sword at the dragon!
 Quizatzhaderac
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Re: Alternate probabilty models
Will wrote:These two are actually at odds. Essentially, what you're talking about with depletable dice (and Adam's "believable coin flip" technique is the same principle) is making a random process conform to the Gambler's Fallacy. This will obviously make streaks less common in general, since the probability of some event A is reduced when the previous random process chose A.
Oh, I didn't mean to say they weren't at odds. One certainly shouldn't use all of these mechanics at all time in all games. One should pick which ones are practically implementable and appropriate to the nature of the game. Though if we wanted both we'd probably do something like Conman's example where inverse gambler's fallacy is true over small samples, and the gambler's fallacy is true over long samples.
Adam H wrote:Yeah, as far as streaks go, I'm not sure there's any way to model what the typical moron thinks.
Sure there is, morons are just somewhat harder to model. The moron is himself an (implicit) model of what he thinks. The problem is that morons disagree, in which case one must either pick a moron, or average them somehow.
Shivahn wrote:I eat a sandwich! I swing at the orc! I spit on the ground! I climb the mountain! I look at Dwayne! I throw the sword at the dragon!
Assuming you're not using a computer to track your streaks based upon a full and accurate model of the probability mechanism: it should be pretty hard to tell if you have the {1, 6, 1, 6, 1, 6} streak or the {6,1,6,1,6,1} streak coming up. With this (or other gambler's fallacy mechanics) I guess you could go around looking at Dwayne until you can expect better rolls, but that should in essence require geometrically more rolls (and time) then you can expect to get beneficial result out for at the end.
The game could also remove or eliminate mundane rolls. Either eating a sandwich is guaranteed, or you can eat two before being full; after awhile Dwayne thinks you're flirting with him, etcetera.
Also separate dice could be used for combat and whatever category eating, spitting, and Dwayne gazing fall into.
The thing about recursion problems is that they tend to contain other recursion problems.
Re: Alternate probabilty models
ConMan wrote:Combining streaks and depletion (or at least a certain amount of memory) could be achieved, though, by having a pool of streaks to choose from rather than individual rolls. So once you've used up the {1, 2, 3, 1, 1, 4} streak, you then might hit the {5, 6, 6, 6, 4, 6} one. Or maybe the {1, 6, 1, 6, 1, 6} one, for swingy luck.
My preferred method would be to have a transition region where, depending on the length of the streak the bias changes. For small streaks, the gambler's fallacy would reign but for larger ones the streak would have momentum.
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Re: Alternate probabilty models
Quizatzhaderac wrote:after awhile Dwayne thinks you're flirting with him, etcetera.
I fail to see the problem.
My plan even makes the bad rolls good!
Re: Alternate probabilty models
Theres also the belief that something that happens with probability 1/100 will happen exactly once per 100 events, for sure. Theres the varient of whether that's every 100 events regardless of if the 'player' is actually participating, or if it's 100 events specific to that person. Either one of these is easily implemented and is basicly equivalent to stuff that's been discussed already.
I was also recently in a discussion where people were insisting that 100/300 odds were much better odds than 1/3, since you had more chances for success. They agreed with me that the probability was the same, and yet the first one was somehow still better. I don't know how you would even begin to model that one in a way that makes them right while keeping the probability the same.
I was also recently in a discussion where people were insisting that 100/300 odds were much better odds than 1/3, since you had more chances for success. They agreed with me that the probability was the same, and yet the first one was somehow still better. I don't know how you would even begin to model that one in a way that makes them right while keeping the probability the same.
 Menacing Spike
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Re: Alternate probabilty models
HoMMV did this with ghosts
they had 50% chance to evade single target hits
if they evaded twice the next attack would hit
if they were hit twice the next attack would be evaded
they had 50% chance to evade single target hits
if they evaded twice the next attack would hit
if they were hit twice the next attack would be evaded
 Will
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Re: Alternate probabilty models
Dopefish wrote:I was also recently in a discussion where people were insisting that 100/300 odds were much better odds than 1/3, since you had more chances for success. They agreed with me that the probability was the same, and yet the first one was somehow still better. I don't know how you would even begin to model that one in a way that makes them right while keeping the probability the same.
I think they're describing sample size without really understanding it. I.e., if you roll a d3 3 times, the probability that it will come up 1 once is smaller than the probability of getting 100 1's in 300 rolls.
That has fuckall to do with odds, though. Odds and probability are the exact same thing, expressed differently.
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Re: Alternate probabilty models
Will wrote:Dopefish wrote:I was also recently in a discussion where people were insisting that 100/300 odds were much better odds than 1/3, since you had more chances for success. They agreed with me that the probability was the same, and yet the first one was somehow still better. I don't know how you would even begin to model that one in a way that makes them right while keeping the probability the same.
I think they're describing sample size without really understanding it. I.e., if you roll a d3 3 times, the probability that it will come up 1 once is smaller than the probability of getting 100 1's in 300 rolls.
That has fuckall to do with odds, though. Odds and probability are the exact same thing, expressed differently.
That's not really correct, either. I think it might be more of a feeling that you have a higher chance of getting _close_ to 100 1's in 300 rolls than you have to getting one 1 in 3 rolls: 85 1's in 300 rolls still feels close to 100, but no 1's in 3 rolls feels like a bigish deal. The more rolls, the more continuous the math.
 Quizatzhaderac
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Re: Alternate probabilty models
With the 1/3 versus 100/300 thing:
They both have the same average, but they have a different standard deviation. Which is (as Will pointed out) is a result of the sample size. If you consider the "result" to be both the average and the standard deviation, 100 D3 /100 is different than D3.
People (for some things) like a more certain result. For instance for farming a dragon, they're okay with having to kill 5 dragons for a 50% chance of fancy pants of awesomeness, but would be really unhappy with having to kill 17 for a 90% chance as that screws up naive budgeting. And even if you do know statistics, that doesn't mean you actually want to use it while playing a game.
For things like crits, people usually like larger variation and a large negative skew, where all of the dramatic results are positive.
They both have the same average, but they have a different standard deviation. Which is (as Will pointed out) is a result of the sample size. If you consider the "result" to be both the average and the standard deviation, 100 D3 /100 is different than D3.
People (for some things) like a more certain result. For instance for farming a dragon, they're okay with having to kill 5 dragons for a 50% chance of fancy pants of awesomeness, but would be really unhappy with having to kill 17 for a 90% chance as that screws up naive budgeting. And even if you do know statistics, that doesn't mean you actually want to use it while playing a game.
For things like crits, people usually like larger variation and a large negative skew, where all of the dramatic results are positive.
The thing about recursion problems is that they tend to contain other recursion problems.
Re: Alternate probabilty models
Dopefish wrote:I was also recently in a discussion where people were insisting that 100/300 odds were much better odds than 1/3, since you had more chances for success. They agreed with me that the probability was the same, and yet the first one was somehow still better. I don't know how you would even begin to model that one in a way that makes them right while keeping the probability the same.
You should look at how people try to reason with the Monty Hall problem.
I have seen incredible posts stating things like "Well, the probability of the car being behind a switched door is better, but the chances are equal."
Good times.
Re: Alternate probabilty models
Interesting concept...
I think some of these models have already been implemented, actually:
* Doesn't Team Fortress 2 increase your chance to get a crit when you have recently gotten a crit? (This would be the "streak" mechanism.) Or is it just when you're dealing lots of damage? (Either one would work for the Heavy's minigun, since without this mechanism, you'd just get a random crit every so often, adding +10 damage or whatever twice a minigun bullet is, and not having much noticeable effect except in the long term!)
* Cosmic Encounter, among other board games, uses cards (well, actually little discs with colored circles on one side, but they behave like cards) to determine who gets attacked; instead of a random chance to attack any opponent as you'd get with dice, you get the "due results" mechanism where after someone's color has been drawn, it's less likely to come up again until the cards are depleted and reshuffled. Apparently many people feel this is "fairer" than dice... (Though Cosmic Encounter does throw in a little nod to randomness, and doesn't let you use the last card in the stack, since it's entirely predictable as all the other cards are gone!)
I think some of these models have already been implemented, actually:
* Doesn't Team Fortress 2 increase your chance to get a crit when you have recently gotten a crit? (This would be the "streak" mechanism.) Or is it just when you're dealing lots of damage? (Either one would work for the Heavy's minigun, since without this mechanism, you'd just get a random crit every so often, adding +10 damage or whatever twice a minigun bullet is, and not having much noticeable effect except in the long term!)
* Cosmic Encounter, among other board games, uses cards (well, actually little discs with colored circles on one side, but they behave like cards) to determine who gets attacked; instead of a random chance to attack any opponent as you'd get with dice, you get the "due results" mechanism where after someone's color has been drawn, it's less likely to come up again until the cards are depleted and reshuffled. Apparently many people feel this is "fairer" than dice... (Though Cosmic Encounter does throw in a little nod to randomness, and doesn't let you use the last card in the stack, since it's entirely predictable as all the other cards are gone!)
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Re: Alternate probabilty models
I feel like I'd probably enjoy a probability model that looked something like (a+n)/(b+n), where a/b is the original probability for whatever event, and n is the the length of your failing streak.
I'm playing around in my head if that would work out to being effectively equivalent to having an average probability rate of some c/d where a/b<c/d. If so, things could still be balanced to be essentially the original rate through an appropriate choice of a new a'/b', yet players could at least be reassured that their chances are getting slightly better with every kill so they wouldn't feel like they'll never get the drop (or whatever it is).
I'm playing around in my head if that would work out to being effectively equivalent to having an average probability rate of some c/d where a/b<c/d. If so, things could still be balanced to be essentially the original rate through an appropriate choice of a new a'/b', yet players could at least be reassured that their chances are getting slightly better with every kill so they wouldn't feel like they'll never get the drop (or whatever it is).
Re: Alternate probabilty models
Dopefish wrote:yet players could at least be reassured that their chances are getting slightly better with every kill so they wouldn't feel like they'll never get the drop (or whatever it is).
Or worse. If a/b>1/2, then (a+n)/(b+n) will be smaller than a/b for all +ve n.
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 Menacing Spike
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Re: Alternate probabilty models
ekolis wrote:* Doesn't Team Fortress 2 increase your chance to get a crit when you have recently gotten a crit? (This would be the "streak" mechanism.) Or is it just when you're dealing lots of damage? (Either one would work for the Heavy's minigun, since without this mechanism, you'd just get a random crit every so often, adding +10 damage or whatever twice a minigun bullet is, and not having much noticeable effect except in the long term!)
Crit rate increases with recent damage/healing dealt, up to a maximum of about 3* the rate (so about 50% crit rate with a melee, that's why medics always melee crit).
The flamethrower and the minigun and the pistol seem to get "bursts" of continuous crits sometimes, I don't know how this works.
Re: Alternate probabilty models
eSOANEM wrote:Dopefish wrote:yet players could at least be reassured that their chances are getting slightly better with every kill so they wouldn't feel like they'll never get the drop (or whatever it is).
Or worse. If a/b>1/2, then (a+n)/(b+n) will be smaller than a/b for all +ve n.
Eh? Let a=3, b=4, so a/b=.75>1/2. Then with n=1, (a+n)/(b+n)=4/5=.8, which is larger, so that's a counter example.
By my reckoning, for a given n your probability will change by (ba)n/(b*(b+n)), which is strictly positive since b>a and a,b,n>0 since it's a probability.
Re: Alternate probabilty models
ekolis wrote:* Doesn't Team Fortress 2 increase your chance to get a crit when you have recently gotten a crit? (This would be the "streak" mechanism.) Or is it just when you're dealing lots of damage? (Either one would work for the Heavy's minigun, since without this mechanism, you'd just get a random crit every so often, adding +10 damage or whatever twice a minigun bullet is, and not having much noticeable effect except in the long term!)
As Spike mentioned, your crit chance increases with damage dealt in the last 20 seconds (url=http://wiki.teamfortress.com/wiki/File:CritHitChance.png]graph[/url]). Since crits themselves do more damage, this can lead to a bit of streaking. But if you miss your crit you won't get any bonus.
The flamethrower and the minigun and the pistol seem to get "bursts" of continuous crits sometimes, I don't know how this works.
All continuous fire weapons get crits in two second bursts. This includes all flamethrowers, miniguns, pistols (but not revolvers), SMGs, syringe guns, and the Shortstop. This is probably in order to give a similar standard deviation as the single shot weapons: If a rapid fire weapon received crits on a pershot basis, they fire fast enough that the law of large numbers would make the actual DPS more much predictable than singleshot crits.
Re: Alternate probabilty models
Dopefish wrote:eSOANEM wrote:Dopefish wrote:yet players could at least be reassured that their chances are getting slightly better with every kill so they wouldn't feel like they'll never get the drop (or whatever it is).
Or worse. If a/b>1/2, then (a+n)/(b+n) will be smaller than a/b for all +ve n.
Eh? Let a=3, b=4, so a/b=.75>1/2. Then with n=1, (a+n)/(b+n)=4/5=.8, which is larger, so that's a counter example.
By my reckoning, for a given n your probability will change by (ba)n/(b*(b+n)), which is strictly positive since b>a and a,b,n>0 since it's a probability.
Sorry, I messed up the maths, for some reason I thought it'd head to 1/2 but it heads to 1. You were quite right.
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Re: Alternate probabilty models
Argh, I cannot find sourcing for what I'm about to say. Regardless of that, and even if its false, it presents an interesting probability model.
Anyway, some people disassembled portions of the XCOM engine and found out that on normal, the probability is... not what is shown.
(Again, I emphasize that I can't find sourcing, so take this with a grain of salt. The idea's the more important thing.)
The numbers for hitting are basically fudged by a benevolent DM. Each consecutive miss makes the next hit more likely, and each hit resets it. The number of active enemies compared to the number of soldiers left was also calculated, and then beneficially applied to the player (so, the worst possible shots are base case, enemies and humans are all still alive. Probability only gets more friendly.)
I think I'm actually fond of the idea behind this model, even if I'm a masochist and prefer to play classic. It makes it so you can still lose, but at no point are you totally fucked from a single bad roll, and it's unlikely to get into positions where you are going to definitely lose, which will happen in truly fair models when you're outnumbered. This lets there be tension, even if it's false, all the time  you don't really have a foregone conclusion.
Anyway, some people disassembled portions of the XCOM engine and found out that on normal, the probability is... not what is shown.
(Again, I emphasize that I can't find sourcing, so take this with a grain of salt. The idea's the more important thing.)
The numbers for hitting are basically fudged by a benevolent DM. Each consecutive miss makes the next hit more likely, and each hit resets it. The number of active enemies compared to the number of soldiers left was also calculated, and then beneficially applied to the player (so, the worst possible shots are base case, enemies and humans are all still alive. Probability only gets more friendly.)
I think I'm actually fond of the idea behind this model, even if I'm a masochist and prefer to play classic. It makes it so you can still lose, but at no point are you totally fucked from a single bad roll, and it's unlikely to get into positions where you are going to definitely lose, which will happen in truly fair models when you're outnumbered. This lets there be tension, even if it's false, all the time  you don't really have a foregone conclusion.
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