Is there any hope of a Cold Iron Normal Distribution?
Yea, I know, a pretty long shot, a specialized dice mechanic for a very obscure game...
Optionally, any chance for table look ups in the dice roller? We could add tables to campaigns and then the dice roller would roll against them. With that, I could implement a d10,000 table to cover a pretty good part of the Cold Iron normal distribution. Of course it would then be handy to have some way to import/export the tables...
Long shot request - Cold Iron Normal Distribution
Re: Long shot request - Cold Iron Normal Distribution
This is a question for Greg, but it might not be the worst idea to detail what is even involved in this.
I know I have no clue.
Re: Long shot request - Cold Iron Normal Distribution
Yeah I have no idea what that is. Please explain and I'll see if I can implement it.
"Sir, our research shows that the bird is equal to or greater than the word."
Re: Long shot request - Cold Iron Normal Distribution
Long but hopefully complete description:
The Chance Adjustment Chart
Cold Iron makes use of a chart that allows random resolution following the normal distribution, a type of bell curve that is open ended. While many systems use 3d6 or other combinations to get a bell curve, some systems also have some way of open ending the roll (exploding dice or some such). The Cold Iron Chance Adjustment Chart is a mechanism to use the open ended normal distribution in the same way. This presents a smooth curve (most methods of exploding dice create funny shaped probability curves, and some make certain rolls impossible - for example, if you have a 1d20 and it explodes on a 20 by rolling another d20 and adding, you can never roll an actual 20, the rolls go 1, 2, 3,... 19, 21, 22,... 29, 31, …).
The Chance Adjustment Chart is a cumulative probability chart for the normal distribution, for example, on a d20, the cumulative probability of rolling a 20+ is 5%, the cumulative probability of rolling a 19+ is 10% (5% chance of a 20, 5% chance of a 19), etc. Probabilities are expressed as a value between 0 and 1 (multiple by 100 to get percent). To use the normal distribution cumulative probability in a game, Mark Christiansen realized instead of rolling 2 ten sided dice (well, we used 20 sided dice numbered 0-9 twice in those days…) to roll a number from 1-100 (or 0-99), you could use them to roll the first two decimal places of a number between 0 and 1. When you look at a cumulative probability chart for the normal distribution you see it starts off flat, then starts to get steeper, eventually reaching a slope of 1:1 in the middle and then starting to flatten off again. At the tail ends, the steps between evenly distributed points become closer and closer together such that two decimal places would not be enough to distinguish them. So what do you do? You pick up more 10 sided dice and roll more digits… Due to the shape of the chart, a simple rule suffices, if you roll leading 9s, keep rolling until you have two digits past a string of leading 9s, and the same for a leading string of 0s.
Now to make use of this in a game, a simple way is to get some kind of positive or negative number out of it to add to, for example, an attack rating, to be compared to a defensive rating. It makes sense to center the output on +0 since the open ended nature means any value in either direction is possible. This also makes the results symmetrical. Mark decided that making 3 standard deviations be translate to a +20 would make a nice range of numbers. Plugging that into the formulas for the normal distribution results in the following table, the Chance Adjustment Chart:
.000088 -25
.00016 -24
.00028 -23
.00048 -22
.00082 -21
.0013 -20
.0022 -19
.0035 -18
.0054 -17
.0082 -16
.012 -15
.018 -14
.025 -13
.036 -12
.049 -11
.067 -10
.088 -9
.12 -8
.15 -7
.18 -6
.23 -5
.27 -4
.33 -3
.38 -2
.44 -1
.50 0
.56 +1
.62 +2
.67 +3
.73 +4
.77 +5
.82 +6
.85 +7
.88 +8
.912 +9
.933 +10
.951 +11
.964 +12
.975 +13
.982 +14
.988 +15
.9918 +16
.9946 +17
.9965 +18
.9978 +19
.9987 +20
.99918 +21
.99952 +22
.99972 +23
.99984 +24
.999912 +25
The decimal numbers, as a cumulative probability are the number rolled using the ten sided dice you need to equal or exceed to get that modifier to add to your rating.
The math and explanation behind this system are complex, but isn’t really any harder to use than, for example, the Universal Table from Marvel Super Heroes. In play, each player has a reference sheet that has the Chance Adjustment Chart as well as several other useful tables for the game. To make an attack, you roll two ten sided dice and look up the result. In fact, if you can memorize 5 or 6 of the numbers, that covers most of your attack rolls without having to consult the table (this is because the combat system tends to require a +0 or better modifier to succeed so you don’t have to memorize much of the bottom half of the table). And once you start to establish how tough the opponent is, many rolls fall easily into failure or simple success (the open ended chart allows for open ended critical hits, but there’s a space of +7 between simple success and the first level of critical hit). And most of the time, of course, you only have to roll two dice (80% of the time). If you have to roll more than two dice, chances are something exciting is about to happen, either a big crit, an actual hit against a hard to hit opponent, or if you rolled a string of zeros, maybe a fumble (fumbles become quite rare with characters who are beyond beginning characters unless you are saddled with some hefty penalties).
The Chance Adjustment Chart
Cold Iron makes use of a chart that allows random resolution following the normal distribution, a type of bell curve that is open ended. While many systems use 3d6 or other combinations to get a bell curve, some systems also have some way of open ending the roll (exploding dice or some such). The Cold Iron Chance Adjustment Chart is a mechanism to use the open ended normal distribution in the same way. This presents a smooth curve (most methods of exploding dice create funny shaped probability curves, and some make certain rolls impossible - for example, if you have a 1d20 and it explodes on a 20 by rolling another d20 and adding, you can never roll an actual 20, the rolls go 1, 2, 3,... 19, 21, 22,... 29, 31, …).
The Chance Adjustment Chart is a cumulative probability chart for the normal distribution, for example, on a d20, the cumulative probability of rolling a 20+ is 5%, the cumulative probability of rolling a 19+ is 10% (5% chance of a 20, 5% chance of a 19), etc. Probabilities are expressed as a value between 0 and 1 (multiple by 100 to get percent). To use the normal distribution cumulative probability in a game, Mark Christiansen realized instead of rolling 2 ten sided dice (well, we used 20 sided dice numbered 0-9 twice in those days…) to roll a number from 1-100 (or 0-99), you could use them to roll the first two decimal places of a number between 0 and 1. When you look at a cumulative probability chart for the normal distribution you see it starts off flat, then starts to get steeper, eventually reaching a slope of 1:1 in the middle and then starting to flatten off again. At the tail ends, the steps between evenly distributed points become closer and closer together such that two decimal places would not be enough to distinguish them. So what do you do? You pick up more 10 sided dice and roll more digits… Due to the shape of the chart, a simple rule suffices, if you roll leading 9s, keep rolling until you have two digits past a string of leading 9s, and the same for a leading string of 0s.
Now to make use of this in a game, a simple way is to get some kind of positive or negative number out of it to add to, for example, an attack rating, to be compared to a defensive rating. It makes sense to center the output on +0 since the open ended nature means any value in either direction is possible. This also makes the results symmetrical. Mark decided that making 3 standard deviations be translate to a +20 would make a nice range of numbers. Plugging that into the formulas for the normal distribution results in the following table, the Chance Adjustment Chart:
.000088 -25
.00016 -24
.00028 -23
.00048 -22
.00082 -21
.0013 -20
.0022 -19
.0035 -18
.0054 -17
.0082 -16
.012 -15
.018 -14
.025 -13
.036 -12
.049 -11
.067 -10
.088 -9
.12 -8
.15 -7
.18 -6
.23 -5
.27 -4
.33 -3
.38 -2
.44 -1
.50 0
.56 +1
.62 +2
.67 +3
.73 +4
.77 +5
.82 +6
.85 +7
.88 +8
.912 +9
.933 +10
.951 +11
.964 +12
.975 +13
.982 +14
.988 +15
.9918 +16
.9946 +17
.9965 +18
.9978 +19
.9987 +20
.99918 +21
.99952 +22
.99972 +23
.99984 +24
.999912 +25
The decimal numbers, as a cumulative probability are the number rolled using the ten sided dice you need to equal or exceed to get that modifier to add to your rating.
The math and explanation behind this system are complex, but isn’t really any harder to use than, for example, the Universal Table from Marvel Super Heroes. In play, each player has a reference sheet that has the Chance Adjustment Chart as well as several other useful tables for the game. To make an attack, you roll two ten sided dice and look up the result. In fact, if you can memorize 5 or 6 of the numbers, that covers most of your attack rolls without having to consult the table (this is because the combat system tends to require a +0 or better modifier to succeed so you don’t have to memorize much of the bottom half of the table). And once you start to establish how tough the opponent is, many rolls fall easily into failure or simple success (the open ended chart allows for open ended critical hits, but there’s a space of +7 between simple success and the first level of critical hit). And most of the time, of course, you only have to roll two dice (80% of the time). If you have to roll more than two dice, chances are something exciting is about to happen, either a big crit, an actual hit against a hard to hit opponent, or if you rolled a string of zeros, maybe a fumble (fumbles become quite rare with characters who are beyond beginning characters unless you are saddled with some hefty penalties).
Re: Long shot request - Cold Iron Normal Distribution
One way to handle the above is to make all the numbers 6 digits by adding zeros and then roll 1d1,000,000. I could give the rest of the 6 digit entries. Of course if the random generator isn’t good enough to do d-million in one go you could do 3 1d100 rolls and tack them together. They would be 0-99 not 1-100 or 0-999,999 for the d-million.
Re: Long shot request - Cold Iron Normal Distribution
Hmm, to revive this idea... Is there any room for outside contributions to the die roller? It would be really cool to implement this within the die roller and maybe I could do some of the coding work...