Thread: Distribution of Chaos Scroll Results

1. Distribution of Chaos Scroll Results

I wasn't getting much reception on basil (lol) so I figured I'd repost it here. I apologize if this is the wrong section, I made my account about 5 minutes ago. This is the original thread

Introduction

Having gotten my 5 year old character to level 200 I'm ready to leave Maplestory behind and move on. However, before I leave I'd like to answer a question that I've pondered over for a while: How are chaos scroll results determined? Most people assume that chaos scroll results are uniform; that there is an equal probability of obtaining each change. However, my experience using chaos scrolls leads me to believe that the results may be approximately normally distributed; the chance of getting a +1 is greater than the chance of getting a +2, +2 greater than +3, ect.

Probability Distributions

The two distributions that I will be focusing on in this experiment are the uniform distribution and the normal distribution.

The uniform distribution is exactly what it sounds like; every outcome has the same probability of occurring. The rolling of a die is an example of a uniform distribution; each outcome of the die has an equal probability of occurring (1/6). If you roll the die many times, you will find that each result appeared an approximately equal number of times. The outcomes will be distributed approximately uniformly link.

The normal distribution is a distribution in which the probability of an event decreases as you further deviate from an expected result. For this you can picture the rolling of two dice and taking their sum (assuming each die is independent, which is a safe assumption). There is only one way to get a sum of 2 (1 & 1) or 12 (6 & 6), but multiple ways to get a sum of 3 (1 & 2|2 &1) , a sum of 7 (1&6|2&5|3&4|4&3|5&2|6&1 ). Therefore, you'd expect to get a sum of 7 more than you'd expect to get a sum of 2 or a sum of 12. Here is the theoretical distribution of summing dice results; each probability is determined by dividing the possible outcomes for each event by the total number out outcomes. For a sum of 2 this is 1/36; for a sum of 7 this is 6/36. If you were to roll a die many times you'd expect the results to resemble this. Notice that the distribution is symmetrical; the probability of getting an 8 is the same as the probability of getting a 6. Also notice the bell-like shape of the distribution. This is characteristic of normal distributions and is known as a Bell Curve.

Chaos Scrolls

Chaos scrolls have the ability to make each stat better or worse by a maximum of 5 points. The possible results of a chaos scroll on a stat are: +5, +4, +3, +2, +1, 0, -1, -2, -3, -4, -5 (for hp/mp multiply each number by 10). If chaos scroll results are normally distributed then each outcome has a probability of 1/11 of happening. If the results are normally distributed then each outcome has an unknown probability. No change would be the most likely, +-1 would be the next most likely, ect.

Assumptions

-Each stat on an item is independent from other stats
-Chaos scroll results are independent from each other
-All stats change in the same way (int and mdef have the same probability distribution)
-Chaos scroll changes do not depend on the quantity of a stat on an item as long as the stat exists on the item (a+10 stat changes with the same probability distribution as a +180 stat)

Requirements

-The chaos scroll must be successful
-A stat must be 5 or greater before the chaos scroll(50 or greater for hp/mp) for the result to be recorded. If a stat is less than 5 and a chaos scroll takes the stat away, then I am left with an ambiguous result (was it -5 or -4?). To eliminate this ambiguity, I will simply ignore the change to a stat that is 4 or less and consider this stat invalid for the current chaos scroll.

Method

-I will buy many chaos scrolls, hopefully well over 100.
-I will chaos items that will give me many valid results. The items that I am currently focusing on are lvl 70 maple hats (10 valid stats), targa helms, scar helms, and zhelms (8 valid stats)
-The results will be recorded

Results

-After 81 valid changes this is the graph that was produced.
I need to wait for some of my stuff to sell before I can do more. This is a work in progress and I will try to update this thread as much as I can until the results are satisfactory.

-After 248 valid changes this is the graph that was produced.
The Bell Curve is becoming apparent; most values are clustered towards the center of the graph. It's almost obvious at this point that the probability of having a stat change by 5 is not the same as the probability of a stat changing by 1. I'd still like to keep going with this; I have more stuff to sell.

If you have any questions or comments, or if you see an error anywhere in my post (a careless definition, spelling error, ect), please let me know. If you'd like to contribute then please post your raw data in the comments.

2. Can you do any testing with Miraculous/Incredible Chaos to see if it keeps the same equation, or if it simply gives the same value to +5 as +12 or +10, and scaling it?

3. I'm a bit limited on funds but I'll see what I can do. I locked most of my equips due to the hacking scare, so there are only a few things I can sell.

4. Making Legend character mules would work for this, a Miraculous Chaos is the reward for 3 awards, which you can get pretty easily.

5. Could you try plotting it against a binomial distribution with n = 10, p = 0.55, minus 5 from the outcome? (so possible values range from -5 to +5) Seems like the most probable distribution to me given the results you've shown so far.

Actually, it would be better to do a best-fit for the 'p' value. 0.6 is just what I eyeballed from the plot.

The set of probabilities with n=10, p=0.55 is:
[1] 0.0003405063 0.0041617435 0.0228895894 0.0746031063 0.1595677552
[6] 0.2340327076 0.2383666466 0.1664782929 0.0763025509 0.0207241496
[11] 0.0025329516
That's -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5.

Actually, looking at it the tails are much too heavy. Multiplying for 248 tests gives 0.1, 1, 6, 19, 40, 58, 59, 41, 19, 5, 1 as the numbers of each, approximately.

6. To be honest I'm a bit rusty when it comes to probability. I just wanted to bring this to peoples' attention, because it seemed like something being completely ignored.
Here's my raw data, I'm sure most of you can do much more with it than I can:
+5(4)
+4(4)
+3(27)
+2(34)
+1(52)
0(46)
-1(30)
-2(24)
-3(9)
-4(10)
-5(8)
I still have funds to blow, I can keep using normal chaos scrolls or move on to miraculous.

7.

8. My results don't match up with anything in that thread, so I'll continue with normal chaos scrolls.

9. Science! I think I have a few Chaos Scrolls lying around somewhere if you want them to use, I'll have to check a bit later. I'd guess all types of chaos scroll work in a similar way, doesn't really make sense to change it if the only difference is stats added/taken.

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