Conduit

Re: The Last Word Don't do that. Instead invest in a natural gas continuous water heater. Rather than heating water and storing it on the off chance you might use it, it doesn't heat water until you turn hot water on. After a few seconds, it delivers hot water and will continue to deliver hot water until you turn off the hot water. It cannot run out of hot water because it heats on demand.

Re: The Last Word What do want from me? I'm an accountant, not an English teacher.

Re: The Last Word I had my 5th earlier this year. We like to go to the Melting Pot for our anniversary because it's the only time we can justify dropping over $100 on one dinner.

Re: The Last Word Congradulations

Re: The Last Word

Re: The Last Word You kidding? I still do that.

Re: [Campaign] DEFENDERS CONGREGATE!! They give you down time?

Re: The probability and predictability of dice.

Re: The probability and predictability of dice. A roll of 2d6 has a standard deviation of 2.42 and a mean of 7.0. 68.2% of the time the roll will be within 7.0 + or - 34.5%. A roll of 10d6 has a standard deviation of 5.40 and a mean of 35.0. 68.2% of the time the roll will be within 35.0 + or - 15.4%. 34.5%/15.4 = 2.24 10d6 is almost 2 and 1/4 times more predictible than 2d6. Edit: removed extra php code

Re: The probability and predictability of dice.

Re: The probability and predictability of dice.

Re: The probability and predictability of dice. First off, SteveZilla, great questions. Actually, while the bell curve does grow wider, it grows taller as well (speaking of the curve with possible results on the x-axis and number of possible combinations on the y-axis). Correct Agreed It's not a fixed number. 68.2% are within 1, 95% are within 2, 99% are within 3 but that is all that can be said. For example it takes slightly more than 4 to encompass all possible results from 10d6. I'm not sure what you mean here. PhilFleischmann gave us a generalized formula for this. The standard deviation of Nd6 is (35N/12)^0.5. Let me say it this way then, the mean of Xd6 increases significantly faster then the standard deviation. This is where you error in the change of shape of the bell curve fails you. When X becomes large, and for our purposes this happens at 3 but becomes "more true" as the number increases. The number of combinations that form results near the mean increases drastically faster than the number of combinations that form results near the extremes. There will only ever be one combination at each extreme while 4d6 has somewhere in the neighborhood of 44 ways to reach the exact mean. 10d6 while still only having one way to hit each extereme has somewhere north of 100 ways to reach the exact mean. Meanwhile the standard deviation has only increased from 3.4 to 5.4. The statistic that is used to measure this is the coefficient of variation which divides the standard deviation by the mean to give an idea of how large the deviation is The CoV of 4d6 is .24 while 10d6 is .15. The smaller number shows that 10d6 is grouped tighter around the mean than 4d6 is. Same thing. The % chance of rolling a given result is the number of combinations that form that result divided by the total number of possible results. Graphing the number of combinations just saves you the dividing. No, I mean the opposite. 2d6 has a mean of 7, a standard deviation of 2.4 and a coefficient of variation of .34. 2d6+28 has a mean of 35 a standard deviation of 2.4 and a coefficient of variation of .07. Xd6+Y is more predictible than Xd6 because it transfers an absolutely smaller Standard Deviation to a larger mean, making the deviation less significant (a smaller percentage of the result). Xd6+Y is also more predicitble than the number of dice normally thrown at that damage class for the same reason, a smaller than normal standard deviation for that mean The reason is that 5 is a smaller percentage of 35 than it is of 7. This is the point where you have to start adding application. If a result of 7 would be sufficient for what ever I want (maybe an attack against 5 defense) 5 is going to make a big difference. If 35 is sufficient, 5 makes less of a difference.

Re: The probability and predictability of dice.

Re: The probability and predictability of dice. Stupid internet not conveying tone:mad: . Not offended at all, I just didn't really understand you comment. I didn't sample anything I actually ran the numbers on all possible results of 2d6 (and 3d6 I just didn't bother to post them) As far as strange things happening, you are right in as far as the more tests you perform the more likely you are to see a random strange thing. Strange things as a group are not all that rare, it's the specific strange things that are rare. For example I would not be terrible surprised if someone came on the board and said they rolled 4 straight 18s, they would be increadibly unlucky but surely we are as Herodom collected approaching 2 billion rolls it will happen sooner or later. But if you told me you started the night with 4 straight and ended with 4 straight with other results inbetween, I wouldn't believe you.

Re: The probability and predictability of dice. ? Not nearly enough samples? Go ahead and work out a couple more examples for yourself then. I guarantee you that they all fit materially within a normal distrbution. The reason that I did 2d6 as the example is that it is the furthest away from a zero-tolerance normal distribution. By the way the odds of rolling 18 four times in a row on 3d6 are 1 in 2,176,782,336. Despite somewhat robust HERO play I don't think I have come anywhere near 2 billion rolls.

Re: The probability and predictability of dice. I ran the number on 2d6. Total Results:36 Mean: 7 Standard Deviation: 2.45 68.2% of Total Results: 24.6 (25) Mean - 1 StD: 4.55 (5) Mean + 1 StD: 9.45 (9) Number of Results between 5 and 9: 24 Percentage of results within 1 StD: 66.7% I think this is close enough to call a normal distribution at this level of granularity.

Re: The probability and predictability of dice. Some of the quotes on the first page where taken from the orginating thread.

Re: The probability and predictability of dice. I have to disagree. Average is a shorthand term or mean, median or mode. Mean is the sum of a set divided by the number of members in the set. Median is that number which when the members of a set have been arranged in numerical order has the same number of members less than and more than that number. For a set with an odd number of members it will actually be one of the numbers (the 3rd member of a set consisting of 5 members). For a set with an even number of members it will be the mean of the two members (The mean of the 3rd and 4th member in a 6 member set). You mistakenly refer to this as the mean. Mode is that number which occurs most frequently in a set. Given the set [3,5,11,11,18] we have the following result: Mean: 9.6 Median: 11 Mode: 11

Re: The probability and predictability of dice. Mean is one of several methods of averaging (others are median and mode, there may be more but these are the main three). All of these are often referred to as the average of a set of numbers. Mean is the method of summing all of the members of the set and dividing the sum by the number of members. So the mean of 2, 5 , 8, 10 is 6.25. The law of large numbers is, in layman's terms (which is all I really know), as more possible results are added the less effect each result has on the whole. A couple of common examples are batting averages and grade point averages. In the first month of a season, a batter's average can vary wildly, going up and down 100 points or more in a single game. By the end of the season, a single game will only cause a change of less than 5 points, if that. The reason is that each result is much more important to the whole when there are few of them. If I have 3 hits in 10 at bats (a .300 batting average) and go 3 for 3 in a game my average goes to .461. If I have 90 hits in 300 at bats (again .300) and go 3 for 3 in a game my average goes to .306. So in the terms of dice rolling, each additional dice has a much smaller affect on the likely outcome than any of the previous dice rolled, so extreme results (all 6s or all 1s or even anything approaching those results) become less and less likely as the number of dice rolled becomes large. Normal distribution is a situation where several things are true. The first thing that is true is that there are as many possible results greater than the mean as less than the mean. The second thing that is true is that 68.2% of the possible results fall within one standard deviation of the mean and 95% fall within two standard devations of the mean and 99% fall within three standard diviations of the mean. It is the "Bell Curve" which we so often talk about. It is useful because it allows us to make confident predictions about the behaviour over time of the object or circumstance being measured. By rules of probabilty I just mean the calculation of possible results and their chance of happening on a given roll. Correct. 2d6 most extreme result ±5 is in absolute terms equal to the standard deviation of (if I remember correctly) 10d6. But 5 is a much more significant number to 2d6 than it is to 10d6 and completely irrelevent to 1Md6. The standard deviation gets larger as you add dice but becomes less significant to the total faster. Plot on a graph the possible results of a throw of xd6 on the x-axis and the number of ways to make a result on the y-axis. The resultant curve is the distribution curve (and happens to be a normal distribution curve) I mis-typed, I mean to say 2d6+28 Right, it is a different curve because it is shifted to the right. It doesn't vary in any other way.

Re: New Player Hates All The Dice No, no, no. You've gone all sideways on me. 2d6 is going to give a less predictible result than 10d6 will. We know, because of the rules of probabilty, normal distribution and the law of large numbers that 10d6 will, over time, present results much closer to the mean result than 2d6 will. It is, that extent, more predictible. If you change a portion of the 10d6 to a static element and roll the rest you will make it even more predictible. The confusion you have is that 2d6 and 2d6+28 are 2DC and 10DC respectively (Assume an EB if you must). They do not have the same distribution curve or mean. If you define predictibility as the likely hood to roll close to mean, 2d6 will be more predictible because the possible variance is small relative to the mean. The problem is that we never do your statement two. Instead of adding a number to dice, we change dice to a number.

Re: New Player Hates All The Dice Yeah, this is the place where we are talking past one another. Let me see if I can more clearly explain my position. Xd6 and xd6+y have the same distribution curve except that xd6+y is shifted east on the axis by y. I think we all agree on this, and this is what I mean when I said xd6 and xd6+y have the same predictibilty (and I spoke poorly at that point; I should not have said predictibility but distribution). The more dice you roll, the more likely that any two of them will add up to 7. Do I have your agreement with this statement? If so, then I think you can see that as the number of dice rolled increases the chances of having a mean value roll {the number of dice rolled time 3.5 (or the number of pairs of dice rolled times 7)} also increases. This is what Dust Raven and I mean by predictability. Rolling just 2d6 is much more likely to have a relatively large variance from the average roll than 10d6 is likely to have. 10d6 can have large absolute variances but those variance will not have very often. It is more predictible because 68.2% of the rolls will be between 30 and 40 (roughly, I didn't go back and check StD for 10d6). When you substitute 8 set dice and only roll 2, you do infact make it more predictible than actually rolling 10 dice because now 68.2* will be between 33 and 37. But it is the change relative to the total that makes it predictible. Rolling 2d6 always provides a result that is with in plus or minus ~2 of the mean. But when that mean is 7 the difference is much more significant than when it is 35 and that is why we say 2d6 by itself is less predictible. Well, we generally don't deal with specific numbers in HERO, we generally deal with ranges. To steal an example from your next post, a typical range we would deal with is "greater than 40." Beyond that a full standard effect is still considered by me to be rolling 10 dice and yes, it would be a predictible result. 10d6 fully rolled is predictible. It will be between 30 and 40 68.2% of the time. It will be over 40 15.9% of the time. Here's where you contradict yourself. You say the player is interested in his ability to roll an absolute number and then immediately turn around and talk about his ability to roll within a range. The player will look at a fully rolled 10d6 and know they will push damage through, on average, once in every five attempts. Whereas the 2d6+28 player know for sure they never will. You know predictible probable is a bad word for what we mean. All dice rolls of any size follow the same rules consistently over time. In that way they are all equally "predictable." Instead maybe the word I want is reliable. At any rate the point I am trying to make is that as dice rolled gets larger the standard deviation becomes less significant relative to the mean and you can become more assured that you will roll the average roll (after all what significance does a StD of 5 mean to an average of 35 when you need to roll over 40 to get damage through? You know you won't be doing much). Yes, larger absoulte variance become possible but they also become less likely.

Re: New Player Hates All The Dice

Re: To sum up... At this time I would like to point out that the statement you intend for me to pick "45 STUN" is not a statement of absolute variance. It is a statement of result. A statement of absolute variance would be "hits you for 10 more than the mean" which obviously no one says. Why? Because noone care a lick for probability after the event occurs. Probabilty is useful in decision making. Once the decision making is done we deal with reality. Predictible means the same thing whether we are talking about HERO or the chance to draw an inside straight. Your attempt to redefine the word to win the argument is just a logical fallicy.

Re: New Player Hates All The Dice No you're not rolling fewer dice. You "rolled" two of them by setting them to 1 and 6 so your are still rolling 10 dice. This is the crux of our argument. You are not getting more predictabilty by reducing the number of dice rolled. Instead you are increasing predictability by setting a portion of the dice rolled to a predetermined result. a does not equal b. A is 2 DC while b is 10. Therefore b represents more dice than a. Yes, b is more predictable than c but that is because b is c just with some dice set to predetermined results. Relative deviance is important to HERO despite your protestations otherwise. A player uses relative deviance to determine whether or not to even attempt the attack (or at any rate should). If they know there is little chance that a given attack will push through damage because of the likely roll of that attack, they will pick a different attack that is likely to push through damage. That though process is entirely dependant upon relative deviance.

Re: New Player Hates All The Dice If you want to add heat, this is the way to do it. Using terms such as good players and bad players, even in quotes, is periously close to flaming. Now, it's your opinion that the game is better when less dice are used and that's fine. My group just loves grabbing two fist full of dice and letting them fly. Neither of us are wrong. So the question is how does your group find a satisfactory way of rolling less dice? I think any of the methods discribed will work for your group, if everyone in the group uses them and the GM adjusts NPC defenses accordingly. Otherwise, Standard Effect and Partial Standard Effect (xd6+y) just never seem to match up to the full dice roll.