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Sunday, June 5, 2011

Scrolling items in Maplestory. Making the right decision.


Definition of a Scroll Outcome

There are two possible outcomes of a scroll: success or failure. For Dark (Cursed) Scrolls, there is a subdivision of outcomes for the failure outcome: destruction or no destruction. The outcomes of scrolls can be described as statistical events.
Events are independent for scrolls, meaning that the outcome of one scroll does not effect the outcome of any other scroll.

Parameters of Calculations
These calculations and probabilities are based on statistics, and thus we assume inferential conditions when we perform calculations. Therefore, when we come to conclusions using this guide, we must assume that the population, N, is large, among other conditions. Unfortunately, these conditions are not always true for certain scrolls, as they are just too rare. However, for our purposes, insufficient conditions do not have a large effect on our inferences.
Furthermore, there is debate whether or not scrolls in MapleStory are randomized. After all, MapleStory uses computerized randomization, which can never be truly random.
iAstronomy: In real life, this may be the case, but as you all may know, MapleStory is programmed, and thus, made with a computer. Likewise, a computer cannot really be random, because it uses seeds to determine probability, and changes seeds every time the quota is full.

Using the same methods of randomization we assume MapleStory to use, I have written a scrolling simulation program in C and found that as the number of scrolls I used increased, the closer the ratio of successes to scrolls used came to the true ratio, consistent with the Law of Large Numbers. This means that, for our purposes, computer randomization is sufficiently random to use the following statistical analyses.
(PM me for source code or an executable to the command line based scrolling simulator.)

Probabilities of Single Events


The following are percentages (proportions) on scroll successes and failures of 1 event.

100% Scroll
Probability of success, 1 event: 1.0, 100%
Probability of failure, no destruction, 1 event: 0.0, 0%
Probability of failure, destruction, 1 event: 0.0, 0%

90% Scroll - 60% used with Vega's Spell
Probability of success, 1 event: 0.9, 90%
Probability of failure, no destruction, 1 event: 0.1, 10%
Probability of failure, destruction, 1 event: 0.0, 0%

70% Scroll
Probability of success, 1 event: 0.7, 70%
Probability of failure, no destruction, 1 event: 0.15, 15%
Probability of failure, destruction, 1 event: 0.15, 15%

60% Scroll
Probability of success, 1 event: 0.6, 60%
Probability of failure, no destruction, 1 event: 0.4, 40%
Probability of failure, destruction, 1 event: 0.0, 0%

50% Scroll
Probability of success, 1 event: 0.5, 50%
Probability of failure, no destruction, 1 event: 0.50, 50%
Probability of failure, destruction, 1 event: 0.0, 0%

30% Scroll
Probability of success, 1 event: 0.3, 30%
Probability of failure, no destruction, 1 event: 0.35, 35%
Probability of failure, destruction, 1 event: 0.35, 35%

30% Scroll - 10% used with Vega's Spell
Probability of success, 1 event: 0.3, 30%
Probability of failure, no destruction, 1 event: 0.7, 70%
Probability of failure, destruction, 1 event: 0.0, 0%

10% Scroll
Probability of success, 1 event: 0.1, 10%
Probability of failure, no destruction, 1 event: 0.9, 90%
Probability of failure, destruction, 1 event: 0.0, 0%

Apparently, there is some confusion on the destruction probabilities for Dark (Cursed) Scrolls:

"Every time I use a dark scroll, there is a 50% chance that the item will break."
Incorrect. This misconception is probably due to the unclear wording in the description of the scroll. In truth, there is a 50% chance that the item breaks if the scroll fails. For example, a 30% scroll has a 70% chance of failure (1 - 0.3 = 0.7). 50% of the time it fails, which is 70% of the time, the item breaks. Thus we conclude that each time you use a 30% scroll, there is a 35% chance the item will break (0.7 / 2 = 0.35). Likewise, each time you use a 70% scroll, there is a 15% chance the item will break (0.3 / 2 = 0.15).

"Every time my dark scroll fails, my item is destroyed by 50%."
Incorrect. Again, this misconception is due to the unclear wording in the description of the scroll. Item breakage is an event, thus it is caused by one scrolling event. Also see above explanation.

Probabilities of Multiple Events


Multiple Events in Succession
To find out your chances of a certain grouping of events in a row, use the following method.
(probability)^n or (prob.)(prob.)(prob.), etc.

For example:
• What is the probability I will get 3 60% scrolls to work (in a row) ?
0.6 ^ 3, which equals 0.216.
There is a 21.6% chance that 3 consecutive 60% scrolls will work.

• What is the probability that I will get 1 30% and 2 70% scrolls to work (in a row) ?
0.3 x 0.7 ^ 2, which equals 0.147.
There is a 14.7% chance that 1 30% and 2 70% scrolls will work consecutively.

Table of Multiple, Consecutive Events; Success

100% Scroll
It will always work! Go figure.

90% Scroll - 60% used with Vega's Spell
n = Number of scrolls used.
Decimal
Percentual

70% Scroll
1 consecutive success: 0.7, 70%
2 consecutive success: 0.49, 49%
3 consecutive success: 0.343, 34.3%
4 consecutive success: 0.2401, 24.01%
5 consecutive success: 0.16807, 16.807%
6 consecutive success: 0.117649, 11.7649%
7 consecutive success: 0.0823543, 8.23543%
8 consecutive success: 0.05764801, 5.764801%
9 consecutive success: 0.040353607, 4.0353607%
10 consecutive success: 0.0282475249, 2.82475249%

60% Scroll
1 consecutive success: 0.6, 60%
2 consecutive success: 0.36, 36%
3 consecutive success: 0.216, 21.6%
4 consecutive success: 0.1296, 12.96%
5 consecutive success: 0.07776, 7.776%
6 consecutive success: 0.046656, 4.6656%
7 consecutive success: 0.0279936, 2.79936%
8 consecutive success: 0.01679616, 1.679616%
9 consecutive success: 0.010077696, 1.0077696%
10 consecutive success: 0.0060466176, 0.60466176%

50% Scroll
n = Number of scrolls used.
Decimal
Percentual

30% Scroll
1 consecutive success: 0.3, 30%
2 consecutive success: 0.09, 9%
3 consecutive success: 0.027, 2.7%
4 consecutive success: 0.0081, 0.81%
5 consecutive success: 0.00243, 0.243%
6 consecutive success: 0.000729, 0.0729%
7 consecutive success: 0.0002187, 0.02187%
8 consecutive success: 0.00006561, 0.006561%
9 consecutive success: 0.000019683, 0.0019683%
10 consecutive success: 0.0000059049, 0.00059049%

20% Scroll
n = Number of scrolls used.
Decimal
Percentual

10% Scroll
1 consecutive success: 0.1, 10%
2 consecutive success: 0.01, 1%
3 consecutive success: 0.001, 0.1%
4 consecutive success: 0.0001, 0.01%
5 consecutive success: 0.00001, 0.001%
6 consecutive success: 0.000001, 0.0001%
7 consecutive success: 0.0000001, 0.00001%
8 consecutive success: 0.00000001, 0.000001%
9 consecutive success: 0.000000001, 0.0000001%
10 consecutive success: 0.0000000001, 0.00000001% (Yikes!)

5% Scroll
n = Number of scrolls used.
Decimal
Percentual

3% Scroll
n = Number of scrolls used.
Decimal
Percentual

1% Scroll
n = Number of scrolls used.
Decimal
Percentual

Table of Multiple, Consecutive Events; Failure

100% Scroll
It will never fail! Go figure.

90% Scroll - 60% used with Vega's Spell
n = Number of scrolls used.
Decimal
Percentual

70% Scroll
1 consecutive failure: 0.3, 30%
2 consecutive failure: 0.09, 9%
3 consecutive failure: 0.027, 2.7%
4 consecutive failure: 0.0081, 0.81%
5 consecutive failure: 0.00243, 0.243%
6 consecutive failure: 0.000729, 0.0729%
7 consecutive failure: 0.0002187, 0.02187%
8 consecutive failure: 0.00006561, 0.006561%
9 consecutive failure: 0.000019683, 0.0019683%
10 consecutive failure: 0.0000059049, 0.00059049%

60% Scroll
1 consecutive failure: 0.4, 40%
2 consecutive failure: 0.16, 16%
3 consecutive failure: 0.064, 6.4%
4 consecutive failure: 0.0256, 2.56%
5 consecutive failure: 0.01024, 1.024%
6 consecutive failure: 0.004096, 0.4096%
7 consecutive failure: 0.0016384, 0.16384%
8 consecutive failure: 0.00065536, 0.065536%
9 consecutive failure: 0.000262144, 0.0262144%
10 consecutive failure: 0.0001048576, 0.01048576%

30% Scroll
1 consecutive failure: 0.7, 70%
2 consecutive failure: 0.49, 49%
3 consecutive failure: 0.343, 34.3%
4 consecutive failure: 0.2401, 24.01%
5 consecutive failure: 0.16807, 16.807%
6 consecutive failure: 0.117649, 11.7649%
7 consecutive failure: 0.0823543, 8.23543%
8 consecutive failure: 0.05764801, 5.764801%
9 consecutive failure: 0.040353607, 4.0353607%
10 consecutive failure: 0.0282475249, 2.82475249%

20% Scroll
n = Number of scrolls used.
Decimal
Percentual

10% Scroll
1 consecutive failure: 0.9, 90%
2 consecutive failure: 0.81, 81%
3 consecutive failure: 0.729, 72.9%
4 consecutive failure: 0.6561, 65.61%
5 consecutive failure: 0.59049, 59.049%
6 consecutive failure: 0.531441, 53.1441%
7 consecutive failure: 0.4782969, 47.82969%
8 consecutive failure: 0.43046721, 43.046721%
9 consecutive failure: 0.387420489, 38.7420489%
10 consecutive failure: 0.3486784401, 34.86784401%

5% Scroll
n = Number of scrolls used.
Decimal
Percentual

3% Scroll
n = Number of scrolls used.
Decimal
Percentual

1% Scroll
n = Number of scrolls used.
Decimal
Percentual

Multiple Events not in Succession
This is the most important statistical tool in determining scroll outcomes over a span of multiple scrolls. From this, we can determine the probability of x number of successes or failures out of n total, the expected outcome, and the expected yield. The equation to find this is complicated, and for one thing, I can't write it with correct notation here on the Basil forums because of the lack of a mathematics key set. However, if you own a TI-83 or higher model calculator, you can find these statistics using binompdf(n,prob.,x) (Credits to yoshidude65)

To find out your chances of a number of successes or fails from a group of scrolls, use the following method.
[n! / ((n-k)!k!)] x p^k x (1-p)^(n-k); whereas n represents the total scrolls used, k represents the number of successes or failures being questioned, and p represents the probability of the success or failure.

For example:
• What is the probability I will get 3 60% scrolls to work out of 5 total 60% scrolls?
[5! / ((5-3)!3!)] x (0.6)^3 x (0.4)^2, or binompdf(5,0.6,3) on your TI-83+ calculator.
There is a 34.56% chance that 3 60% scrolls will work out of 5 total 60% scrolls.

• What is the probability that I will get 1 10% scroll to work out of 7 total 10% scrolls?
[7! / ((7-1)!1!)] x (0.1)^1 x (0.9)^6, or binompdf(7,0.1,1) on your TI-83+ calculator.
There is a 37.2% chance that 1 10% scroll will work out of 7 total 10% scrolls.

Table of Multiple Events, k successes out of n trials

I realize that only some of the scroll percent types are listed below - they are the most popular. If you would like to find the probability of another set of occurrences, please use this tool.

70% Scroll
1 success out of 2 trials: 0.42, 42%
1 success out of 3 trials: 0.189, 18.9%
1 success out of 4 trials: 0.0756, 7.56%
1 success out of 5 trials: 0.02835, 2.835%
1 success out of 6 trials: 0.010206, 1.0206%
1 success out of 7 trials: 0.0035721, 0.35721%
1 success out of 8 trials: 0.00122472, 0.122472%
1 success out of 9 trials: 0.000413343, 0.0413343%
1 success out of 10 trials: 0.000137781, 0.0137781%
2 success out of 3 trials: 0.441, 44.1%
2 success out of 4 trials: 0.2646, 26.46%
2 success out of 5 trials: 0.1323, 13.23%
2 success out of 6 trials: 0.059535, 5.9535%
2 success out of 7 trials: 0.0250047, 2.50047%
2 success out of 8 trials: 0.01000188, 1.000188%
2 success out of 9 trials: 0.003857868, 0.3857868%
2 success out of 10 trials: 0.0014467005, 0.14467005%
3 success out of 4 trials: 0.4116, 41.16%
3 success out of 5 trials: 0.3087, 30.87%
3 success out of 6 trials: 0.18522, 18.522%
3 success out of 7 trials: 0.0972405, 9.72405%
3 success out of 8 trials: 0.04667544, 4.667544%
3 success out of 9 trials: 0.021003948, 2.1003948%
3 success out of 10 trials: 0.009001692, 0.9001692%
4 success out of 5 trials: 0.36015, 36.015%
4 success out of 6 trials: 0.324135, 32.4135%
4 success out of 7 trials: 0.2268945, 22.68945%
4 success out of 8 trials: 0.1361367, 13.61367%
4 success out of 9 trials: 0.073513818, 7.3513818%
4 success out of 10 trials: 0.036756909, 3.675690%
5 success out of 6 trials: 0.302526, 30.2526%
5 success out of 7 trials: 0.3176523, 31.76523%
5 success out of 8 trials: 0.25412184, 25.412184%
5 success out of 9 trials: 0.171532242, 17.1532242%
5 success out of 10 trials: 0.1029193542, 10.29193542%
6 success out of 7 trials: 0.2470629, 24.70629%
6 success out of 8 trials: 0.29647548, 29.647548%
6 success out of 9 trials: 0.266827932, 26.6827932%
6 success out of 10 trials: 0.200120949, 20.0120949%
7 success out of 8 trials: 0.19765032, 19.765032%
7 success out of 9 trials: 0.266827932, 26.6827932%
7 success out of 10 trials: 0.266827932, 26.6827932%
8 success out of 9 trials: 0.155649627, 15.5649627%
8 success out of 10 trials: 0.2334744405, 23.34744405%
9 success out of 10 trials: 0.121060821, 12.1060821%

60% Scroll
1 success out of 2 trials: 0.48, 48%
1 success out of 3 trials: 0.288, 28.8%
1 success out of 4 trials: 0.1536, 15.36%
1 success out of 5 trials: 0.0768, 7.68%
1 success out of 6 trials: 0.036864, 3.6864%
1 success out of 7 trials: 0.0172032, 1.72032%
1 success out of 8 trials: 0.00786432, 0.786432%
1 success out of 9 trials: 0.003538944, 0.3538944%
1 success out of 10 trials: 0.001572864, 0.1572864%
2 success out of 3 trials: 0.432, 43.2%
2 success out of 4 trials: 0.3456, 34.56%
2 success out of 5 trials: 0.2304, 23.04%
2 success out of 6 trials: 0.13824, 13.824%
2 success out of 7 trials: 0.0774144, 7.74144%
2 success out of 8 trials: 0.04128768, 4.128768%
2 success out of 9 trials: 0.021233664, 2.1233664%
2 success out of 10 trials: 0.010616832, 1.0616832%
3 success out of 4 trials: 0.3456, 34.56%
3 success out of 5 trials: 0.3456, 34.56%
3 success out of 6 trials: 0.27648, 27.648%
3 success out of 7 trials: 0.193536, 19.3536%
3 success out of 8 trials: 0.12386304, 12.386304%
3 success out of 9 trials: 0.074317824, 7.4317824%
3 success out of 10 trials: 0.042467328, 4.2467328%
4 success out of 5 trials: 0.2592, 25.92%
4 success out of 6 trials: 0.31104, 31.104%
4 success out of 7 trials: 0.290304, 29.0304%
4 success out of 8 trials: 0.2322432, 23.22432%
4 success out of 9 trials: 0.167215104, 16.7215104%
4 success out of 10 trials: 0.111476736, 11.1476736%
5 success out of 6 trials: 0.186624, 18.6624%
5 success out of 7 trials: 0.2612736, 26.12736%
5 success out of 8 trials: 0.27869184, 27.869184%
5 success out of 9 trials: 0.250822656, 25.0822656%
5 success out of 10 trials: 0.2006581248, 20.06581248%
6 success out of 7 trials: 0.1306368, 13.06368%
6 success out of 8 trials: 0.20901888, 20.901888%
6 success out of 9 trials: 0.250822656, 25.0822656%
6 success out of 10 trials: 0.250822656, 25.0822656%
7 success out of 8 trials: 0.08957952, 8.957952%
7 success out of 9 trials: 0.161243136, 16.1243136%
7 success out of 10 trials: 0.214990848, 21.4990848%
8 success out of 9 trials: 0.060466176, 6.0466176%
8 success out of 10 trials: 0.120932352, 12.0932352%
9 success out of 10 trials: 0.040310784, 4.0310784%

30% Scroll
1 success out of 2 trials: 0.42, 42%
1 success out of 3 trials: 0.441, 44.1%
1 success out of 4 trials: 0.4116, 41.16%
1 success out of 5 trials: 0.36015, 36.015%
1 success out of 6 trials: 0.302526, 30.2526%
1 success out of 7 trials: 0.2470629, 24.70629%
1 success out of 8 trials: 0.19765032, 19.765032%
1 success out of 9 trials: 0.155649627, 15.5649627%
1 success out of 10 trials: 0.121060821, 12.1060821%
2 success out of 3 trials: 0.189, 18.9%
2 success out of 4 trials: 0.2646, 26.46%
2 success out of 5 trials: 0.3087, 30.87%
2 success out of 6 trials: 0.324135, 32.4135%
2 success out of 7 trials: 0.3176523, 31.76523%
2 success out of 8 trials: 0.29647548, 29.647548%
2 success out of 9 trials: 0.266827932, 26.6827932%
2 success out of 10 trials: 0.2334744405, 23.34744405%
3 success out of 4 trials: 0.0759, 7.59%
3 success out of 5 trials: 0.1323, 13.23%
3 success out of 6 trials: 0.18522, 18.522%
3 success out of 7 trials: 0.2268945, 22.68945%
3 success out of 8 trials: 0.25412184, 25.412184%
3 success out of 9 trials: 0.266827932, 26.6827932%
3 success out of 10 trials: 0.266827932, 26.6827932%
4 success out of 5 trials: 0.02835, 2.835%
4 success out of 6 trials: 0.059535, 5.9535%
4 success out of 7 trials: 0.0972405, 9.72405%
4 success out of 8 trials: 0.1361367, 13.61367%
4 success out of 9 trials: 0.171532242, 17.1532242%
4 success out of 10 trials: 0.200120949, 20.0120949%
5 success out of 6 trials: 0.010206, 1.0206%
5 success out of 7 trials: 0.0250047, 2.50047%
5 success out of 8 trials: 0.04667544, 4.667544%
5 success out of 9 trials: 0.073513818, 7.3513818%
5 success out of 10 trials: 0.1029193452, 10.29193452%
6 success out of 7 trials: 0.0035721, 0.35721%
6 success out of 8 trials: 0.01000188, 1.000188%
6 success out of 9 trials: 0.021003948, 2.1003948%
6 success out of 10 trials: 0.036756909, 3.6756909%
7 success out of 8 trials: 0.00122472, 0.122472%
7 success out of 9 trials: 0.003857868, 0.3857868%
7 success out of 10 trials: 0.009001692, 0.9001692%
8 success out of 9 trials: 0.000413343, 0.0413343%
8 success out of 10 trials: 0.0014467005, 0.14467005%
9 success out of 10 trials: 0.000137781, 0.0137781%

10% Scroll
1 success out of 2 trials: 0.18, 18%
1 success out of 3 trials: 0.243, 24.3%
1 success out of 4 trials: 0.2916, 29.16%
1 success out of 5 trials: 0.32805, 32.805%
1 success out of 6 trials: 0.354294, 35.4294%
1 success out of 7 trials: 0.3720087, 37.20087%
1 success out of 8 trials: 0.38263752, 38.263752%
1 success out of 9 trials: 0.387420489, 38.7420489%
1 success out of 10 trials: 0.387420489, 38.7420489%
2 success out of 3 trials: 0.027, 2.7%
2 success out of 4 trials: 0.0486, 4.86%
2 success out of 5 trials: 0.0729, 7.29%
2 success out of 6 trials: 0.098415, 9.8415%
2 success out of 7 trials: 0.1240029, 12.40029%
2 success out of 8 trials: 0.14880348, 14.880348%
2 success out of 9 trials: 0.172186884, 17.2186884%
2 success out of 10 trials: 0.1937102445, 19.37102445%
3 success out of 4 trials: 0.0036, 0.36%
3 success out of 5 trials: 0.0081, 0.81%
3 success out of 6 trials: 0.01458, 1.458%
3 success out of 7 trials: 0.0229635, 2.29635%
3 success out of 8 trials: 0.03306744, 3.306744%
3 success out of 9 trials: 0.044641044, 4.4641044%
3 success out of 10 trials: 0.057395628, 5.7395628%
4 success out of 5 trials: 0.00045, 0.045%
4 success out of 6 trials: 0.001215, 0.1215%
4 success out of 7 trials: 0.0025515, 0.25515%
4 success out of 8 trials: 0.0045927, 0.45927%
4 success out of 9 trials: 0.007440174, 0.7440174%
4 success out of 10 trials: 0.011160261, 1.1160261%
5 success out of 6 trials: 0.000054, 0.0054%
5 success out of 7 trials: 0.0001701, 0.01701%
5 success out of 8 trials: 0.00040824, 0.040824%
5 success out of 9 trials: 0.000826686, 0.0826686%
5 success out of 10 trials: 0.0014880348, 0.14880348%
6 success out of 7 trials: 0.0000063, 0.00063%
6 success out of 8 trials: 0.00002268, 0.002268%
6 success out of 9 trials: 0.000061236, 0.0061236%
6 success out of 10 trials: 0.000137781, 0.0137781%
7 success out of 8 trials: 0.00000072, 0.000072%
7 success out of 9 trials: 0.000002916, 0.0002916%
7 success out of 10 trials: 0.000008748, 0.0008748%
8 success out of 9 trials: 0.000000081, 0.0000081%
8 success out of 10 trials: 0.0000003645, 0.00003645%
9 success out of 10 trials: 0.000000009, 0.0000009% (Good luck!)


Expected Yield

'Expected value' is synonymous with the word mean, or average. The expected yield of a scroll is the approximate mean outcome of multiple scrolls in terms of payout from the scroll (i.e. Weapon Attack, or DEX).

To find the expected yield of a grouping of scrolls, use the following method.
Find the probability of k successes out of n trials for each possible k. Then multiply each of these probabilities by the payout of the scroll multiplied by k. Add each of these permutations.

For example:
• What's the expected yield from using 5 60% Glove Att. scrolls?
[((5!/0!) x (0.6)^0 x (0.4)^5)0] + [((5!/1!) x (0.6)^1 x (0.4)^4)2] + [((5!/2!) x (0.6)^2 x (0.4)^3)4] + [((5!/3!) x (0.6)^3 x (0.4)^2)6] + [((5!/4!) x (0.6)^4 x (0.4)^1)8] + [((5!/5!) x (0.6)^5 x (0.4)^0)10]
The expected yield of 5 60% Glove Att. scrolls is +6 W. Attack. (Yes all those numbers add up to 6 .)

Table of Expected Yields for 7 Slots; Common Weapon Scrolls
Warning: the expected yield counts for Dark (Cursed) Scrolls do not factor in breakage. Also, these are in no way a guaruntee, nor a limit.

Warrior Weapons
Expected yield of 7 70%: +9.8 Attack, +4.9 STR
Expected yield of 7 60%: +8.4 Attack, +4.2 STR
Expected yield of 7 30%: +10.5 Attack, +6.3 STR, +2.1 W. Def.
Expected yield of 7 10%: +3.5 Attack, +2.1 STR, +0.7 W. Def.

Magician Weapons
Expected yield of 7 70%: +9.8 M. Attack, +4.9 INT
Expected yield of 7 60%: +8.4 M. Attack, +4.2 INT
Expected yield of 7 30%: +10.5 M. Attack, +6.3 INT, +2.1 M. Def.
Expected yield of 7 10%: +3.5 M.Attack, +2.1 INT, +0.7 M. Def.

Bowman Weapons
Expected yield of 7 70%: +9.8 Attack, +4.9 ACC
Expected yield of 7 60%: +8.4 Attack, +4.2 ACC
Expected yield of 7 30%: +10.5 Attack, +6.3 ACC, +2.1 DEX.
Expected yield of 7 10%: +3.5 Attack, +2.1 ACC, +0.7 DEX.

Daggers
Expected yield of 7 70%: +9.8 Attack, +4.9 LUK
Expected yield of 7 60%: +8.4 Attack, +4.2 LUK
Expected yield of 7 30%: +10.5 Attack, +6.3 LUK, +2.1 W. Def.
Expected yield of 7 10%: +3.5 Attack, +2.1 LUK, +0.7 W. Def.

Claws
Expected yield of 7 70%: +9.8 Attack, +4.9 ACC
Expected yield of 7 60%: +8.4 Attack, +4.2 ACC
Expected yield of 7 30%: +10.5 Attack, +6.3 ACC, +2.1 LUK.
Expected yield of 7 10%: +3.5 Attack, +2.1 ACC, +0.7 LUK.

Is scrolling this worth my money?

This is a common question among Maplers, yet many don't know how to go about finding out. With scroll prices through the ceiling (especially Claw Scrolls), it is imperative that you know when to use scrolls, where to use scrolls, and if to use scrolls. How do we go about finding this out? By using expected yield!

When to Scroll
As aforementioned, scrolling takes funding, and placing your hard-earned mesos in the hands of pure chance can be grueling. That being said, here are a few tips on how to know when to scroll:
• Do not attempt scrolling unless you have sufficient funding to prepare for the worst. Say for example I have 14 mil. and I buy 7 60% Staff Scrolls for 2 mil. each. If you end up with only 0-2 working, you would have financial instability. This is why it is important to never engage in scrolling unless you have mesos to spare after the scrolling.
• Attempt scrolling your weapon when you have sufficient funding. The weapon should be the first item upgraded for your set of equipments. They give the highest expected yield for the amount of mesos they cost.
• Attempt scrolling your other equips (determine some order of importance or worth) when you have over-sufficient funding, or in other words, mesos to blow .
• Attempt scrolling when training gets slow - if there's no way to make the experience better from changing griding monsters, you can kill faster with the addition of these wonderful scrolls. Don't limit yourself to just power scrolls, though - Speed, Jump, Avoidability, and Accuracy are all important factors in increasing your killing speed.
• Attempt scrolling when you have over-sufficient funding and you're bored. Seriously, it's pretty fun.

Where to Scroll (and if!)
No, this section doesn't tell you that you should use your scrolls in Channel 6 at some exact location on a Tuesday night from 6 to 7 PM. The objective of this section is to show the reader which equipments to scroll based on their mesos.

After determining that it's time to scroll from a reason in the above section, there is a method of finding which scrolls will give you the best chance of getting the highest yield; use as follows.
Find the expected yields for all of the scroll groupings being questioned, then find the price of each and divide the yields by that price.

For example:
• Which will give me the most DEX per meso: 5 60% Glove DEX, 5 60% Overall DEX, or 5 70% Earring DEX?
Expected yield of 60% Glove DEX: +3 DEX / 500,000 mesos = 0.000006
Expected yield of 60% Overall DEX: +6 DEX / 3,500,000 mesos = 0.000001714
Expected yield of 70% Earring DEX: +6 DEX / 6,000,000 mesos = 0.000001
60% Glove DEX will give about 0.6 DEX per 100,000 mesos,
60% Overall DEX will give about 0.17 DEX per 100,000 mesos,
and 70% Earring DEX will give about 0.1 DEX per 100,000 mesos.

The above method and example were on a fixed n, or number of scrolls used. But what if you have a fixed amount of mesos to spend? Use the following method.
Find the amount of mesos you are using to scroll, then find how many scrolls you can buy using that money - group by similarity. Then use the expected yield calculation. (You won't need to divide by the amount of mesos used unless you don't use all of the mesos set aside for scrolling for a certain group of scrolls)

For example:
• I have 10 mil. to scroll. What will give me the most Magic Attack for my money?
With 10 mil. I can buy 5 60% Earring INT Scrolls, 3 60% Cape INT Scrolls, or 2 60% Overall INT Scrolls.
Expected yield of 5 60% Earring INT: +9 Magic Attack (combined INT and Magic Attack)
Expected yield of 3 60% Cape INT: +3.6 Magic Attack
Expected yield of 2 60% Overall INT: +2.4 Magic Attack
Therefore, from these prices (they will vary), 60% Earring INT will give the most yield for 10 mil.

Disproving Popular Scrolling "Strategies"


Dummy Scrolling
Many people claim to have had astounding luck with this method - so lucky that 'dummy scrolling' must've been a way for them to increase their scrolling probabilities, right? Wrong!
Dummy Scrolling is using cheap scrolls on cheap items with the intention of failing them in order to raise the probability of the following scroll on the real item. Obviously, the maker of this strategy phenomenum forgot that scroll outcomes are INDEPENDENT events, thus the outcome of one individual scroll does not affect the outcome of another. This method is very similar to the Law of Averages that some people quote heartedly, though the Law of Averages actually does not exist and is false! Due to the "Law of Averages," if you toss 6 coins and they all end up heads, your next coin toss is probably going to be a tails - wrong! The toss is still 50% because the events are independent. Likewise, you cannot use dummy scrolling and a combination of the Law of Averages to assume that you can raise your probabilities.

Ritualistic Scrolling
Now, I admit; I use scrolling rituals. They're fun in a way, but in reality: they are false. Lots of people have special scrolling spots, channels, and rituals, but I'm here to break it to you: it doesn't make a bit of difference. Nowhere is exempt to the rules of statistics, thus wherever and whenever you scroll, you always have the same probability. Recall the 'Bob the snail' and Red Skullcap phenomena - those, too, are false for the reason shown above.

Gamblers' Fallacy
I suppose it's a stretch to call this a "strategy," but here we go. Gamblers' Fallacy, which plays hand in hand with the Law of Averages, is when the gambler, or scroller in this case, uses the Law of Averages to assume the next outcomes. For instance, say I used 4 60% Overall LUK Scrolls on my bathrobe and they all failed . Gamblers' Fallacy is assuming that the next one will work because it's due to occur. This is false and by assuming this, you will only be disappointed.

Legendary Spirit (pointed out by Kazoothebat)
Legengary Spirit is a skill attained by a quest that allows people of any job or level to scroll any item. It has come to my attention that some people think that using this skill gives them a greater success rate, however, this is completely false. Though having this skill allows you to scroll, for example, a level 80 thief shoe if you're a level 40 wizard, it will not increase the probability of success of the scroll you are using.

Addendum


White Scrolls

This section is for those of you who are looking to "BSUCLA" your equipment using White Scrolls. I am assuming that if you have this much mesos to go through with this endeavor then you will be using 10% scrolls.

5 Slot Item
(0.1)(10) = 1 success. (10)(5) = 5 success.
The expected number of 10% Scroll / White Scroll combinations that will need to be used to fill up a 5 slot item is 50.

7 Slot Item
(0.1)(10) = 1 success. (10)(7) = 7 success.
The expected number of 10% Scroll / White Scroll combinations that will need to be used to fill up a 7 slot item is 70.

10 Slot Item
(0.1)(10) = 1 success. (10)(10) = 10 success.
The expected number of 10% Scroll / White Scroll combinations that will need to be used to fill up a 10 slot item is 100.

Clean Slate Scrolls

Clean slate scrolls are an addition to the variety of scrolls in Patch 0.56. With a small success rate, they regain lost slots.

Probability of the Clean Slate Scroll as a Single Event
1% Clean Slate Scroll
Probability of success, 1 event: 0.01, 1%
Probability of failure, no destruction, 1 event: 0.97, 97%
Probability of failure, destruction, 1 event: 0.02, 2%

3% Clean Slate Scroll
Probability of success, 1 event: 0.03, 3%
Probability of failure, no destruction, 1 event: 0.91, 91%
Probability of failure, destruction, 1 event: 0.06, 6%

5% Clean Slate Scroll
Probability of success, 1 event: 0.05, 5%
Probability of failure, no destruction, 1 event: 0.85, 85%
Probability of failure, destruction, 1 event: 0.1, 10%

20% Clean Slate Scroll
Probability of success, 1 event: 0.2, 20%
Probability of failure, no destruction, 1 event: 0.4, 40%
Probability of failure, destruction, 1 event: 0.4, 40%

Probability of the Clean Slate Scroll as Multiple Events
1% Clean Slate Scroll
1 consecutive success: 0.01, 1%
2 consecutive success: 0.0001, 0.01%
3 consecutive success: 0.000001, 0.0001%
4 consecutive success: 0.00000001, 0.000001%
5 consecutive success: 0.0000000001, 0.00000001%
6 consecutive success: 0.000000000001, 0.0000000001%
7 consecutive success: 0.00000000000001, 0.000000000001%
8 consecutive success: 0.0000000000000001, 0.00000000000001%
9 consecutive success: 0.000000000000000001, 0.0000000000000001%
10 consecutive success: 0.00000000000000000001, 0.000000000000000001%

The following are probabilities for consecutive failure with NO destruction.
1 consecutive failure: 0.97, 97%
2 consecutive failure: 0.9409, 94.09%
3 consecutive failure: 0.912673, 91.2673%
4 consecutive failure: 0.88529281, 88.529281%
5 consecutive failure: 0.8587340257, 85.87340257%
6 consecutive failure: 0.8329720049, 83.29720049%
7 consecutive failure: 0.8079828448, 80.79828448%
8 consecutive failure: 0.7837433594, 78.37433594%
9 consecutive failure: 0.7602310587, 76.02310587%
10 consecutive failure: 0.7374241269, 73.74241269%

3% Clean Slate Scroll
1 consecutive success: 0.03, 3%
2 consecutive success: 0.0009, 0.09%
3 consecutive success: 0.000027, 0.0027%
4 consecutive success: 0.00000081, 0.000081%
5 consecutive success: 0.0000000243, 0.00000243%
6 consecutive success: 0.000000000729, 0.0000000729%
7 consecutive success: 0.00000000002187, 0.000000002187%
8 consecutive success: 0.0000000000006561, 0.00000000006561%
9 consecutive success: 0.000000000000019683, 0.0000000000019683%
10 consecutive success: 0.00000000000000059049, 0.000000000000059049%

The following are probabilities for consecutive failure with NO destruction.
1 consecutive failure: 0.91, 91%
2 consecutive failure: 0.8281, 82.81%
3 consecutive failure: 0.753571, 75.3571%
4 consecutive failure: 0.68574961, 68.574961%
5 consecutive failure: 0.6240321451, 62.40321451%
6 consecutive failure: 0.567869252, 56.7869252%
7 consecutive failure: 0.5167610194, 51.67610194%
8 consecutive failure: 0.4702525276, 47.02525276%
9 consecutive failure: 0.4279298001, 42.79298001%
10 consecutive failure: 0.3894161181, 38.94161181%

5% Clean Slate Scroll
1 consecutive success: 0.05, 5%
2 consecutive success: 0.0025, 0.25%
3 consecutive success: 0.000125, 0.0125%
4 consecutive success: 0.00000625, 0.00625%
5 consecutive success: 0.0000003125, 0.0003125%
6 consecutive success: 0.000000015625, 0.0000015626%
7 consecutive success: 0.00000000078125, 0.000000078125%
8 consecutive success: 0.0000000000390625, 0.00000000390625%
9 consecutive success: 0.000000000001953125, 0.0000000001953125%
10 consecutive success: 0.00000000000009765625, 0.000000000009765625%

The following are probabilities for consecutive failure with NO destruction.
1 consecutive failure: 0.85, 85%
2 consecutive failure: 0.7225, 72.25%
3 consecutive failure: 0.614125, 61.4125%
4 consecutive failure: 0.52200625, 52.200625%
5 consecutive failure: 0.4437053125, 44.37053125%
6 consecutive failure: 0.3771495156, 37.71495156%
7 consecutive failure: 0.3205770883, 32.05770883%
8 consecutive failure: 0.272490525, 27.2490525%
9 consecutive failure: 0.2316169463, 23.16169463%
10 consecutive failure: 0.1968744043, 19.68744043%

20% Clean Slate Scroll
1 consecutive success: 0.2, 20%
2 consecutive success: 0.04, 4%
3 consecutive success: 0.008, 0.8%
4 consecutive success: 0.0016, 0.16%
5 consecutive success: 0.00032, 0.032%
6 consecutive success: 0.000064, 0.0064%
7 consecutive success: 0.0000128, 0.00128%
8 consecutive success: 0.00000256, 0.000256%
9 consecutive success: 0.000000512, 0.0000512%
10 consecutive success: 0.0000001024, 0.00001024%

The following are probabilities for consecutive failure with or without destruction.
1 consecutive failure: 0.4, 40%
2 consecutive failure: 0.16, 16%
3 consecutive failure: 0.064, 6.4%
4 consecutive failure: 0.0256, 2.56%
5 consecutive failure: 0.01024, 1.024%
6 consecutive failure: 0.004096, 0.4096%
7 consecutive failure: 0.0016384, 0.16384%
8 consecutive failure: 0.00065536, 0.065536%
9 consecutive failure: 0.000262144, 0.0262144%
10 consecutive failure: 0.0001048576, 0.01048576%

I realize that there potentially could be an infinite number of Clean Slate scrolls used on an item since they do not consume a slot. However, I am stopping the tables at 10 consecutive due to the low odds of 10 consecutive successes or failures and due to the sheer length of the table. Nonetheless, it is plausible that someone uses more than 10 Clean Slate scrolls, and in order to find consecutive probabilities for those, as aforementioned in the section entitled 'Multiple Events in Succession,' link, take the probability of the scroll's success and raise it to the power of the number of scrolls you are using.

Chaos Scrolls


Probability of the Chaos Scroll as a Single Event
Probability of success, (Any change), 1 event: 0.6, 60%
Probability of success, Net (overall) Positive change, 1 event: 0.3, 30%
• Probability of success, +5 stat points, 1 event: 0.06, 6%
• Probability of success, +4 stat points, 1 event: 0.06, 6%
• Probability of success, +3 stat points, 1 event: 0.06, 6%
• Probability of success, +2 stat points, 1 event: 0.06, 6%
• Probability of success, +1 stat points, 1 event: 0.06, 6%
Probability of success, Net (overall) Negative change, 1 event: 0.3, 30%
• Probability of success, -5 stat points, 1 event: 0.06, 6%
• Probability of success, -4 stat points, 1 event: 0.06, 6%
• Probability of success, -3 stat points, 1 event: 0.06, 6%
• Probability of success, -2 stat points, 1 event: 0.06, 6%
• Probability of success, -1 stat points, 1 event: 0.06, 6%
Probability of failure, destruction, 1 event: 0.4, 40%

Deviations of Success Effects
The distribution of effects upon scroll success are integers from -5 to 5, excluding 0, which is a failure and subsequent destruction.
Assuming that each deviation has an equal probability of occurrence, each deviation has a probability of 0.06, 6%, meaning that your Chaos Scroll has a 30% chance of adding or subtracting stat points on an overall basis. (See above)

6 comments:

  1. Wow, pretty well researched. Always nice to see the numbers behind these things. :)

    ReplyDelete
  2. awesome research tbh! :D
    nice stuff, keep it up

    ReplyDelete
  3. wow, that must have taken forever!

    ReplyDelete
  4. Good info, i might start playing!

    ReplyDelete
  5. how did you get all that? gee that looks useful!!

    ReplyDelete