GG's Tip-o-the-Week (11-28)

[GooberGrape1]

GG's Tip-o-the-Week (11-28)

The great Meph project is finished, thanks to all who gave their input.

All drops calculated at /players 3 with 500 MF.

First, I assumed mututally exclusive drops. This essentially means that Meph is only capable of dropping 1 Set/Unique per kill. Of course, he drops more often. But the independent probability of any two specific S/U items dropping at a given time is exponentially UN-likely, therefore, I am going to assume it to be zero.

Example:
Odds of Stormlash = 1: 36,321
Odds of Stormlash AND Occulus = 1: 36,248,358

Next, let's try what I believe is a Dynkin system. Let C be a collection of subsets of X. In this example, C would be the entire exceptional grail, and X would be each individual item within C. If D is a Dynkin system containing C, then D also contains each item X within C.

Example:
If Stormlash = 1: 36,321
Then every item that SHOULD drop within 36,321 runs, WILL drop.

Therefore, if Stormlash is the rarest item at 36,321 runs, the minimum number of runs to get every item that Meph can drop is 36,321.


How many Meph runs should I run before it's just not worth it anymore???

To answer that question, I tabulated the individual odds for all the elite uniques in a spreadsheet, and graphed it. The trendline is almost flat, with a huge vertical skew at the end, representing Stormlash and Spirit Keeper.

Removing those two items created a graph definable by fourth order polynomials. I'm not too happy with that, since it still includes a great number of very unlikely drops.

Finally, I graphed 2/3 of the elites (40 of 59). This turned out to be a nice, clean linear equation.
y = 17.874x + 261.93

Therefore, if you are happy with 2/3 of the total elites, you only need to run Meph 1,000 times. Given that Meph cannot drop TC87, I believe this is the most reasonable solution.

Conclusion:

Run Meph 1,000 times, then move on.


GG

 

[shermo]

interesting

I didn't quite understand the relevance of the didkin system though

Example:
If Stormlash = 1: 36,321
Then every item that SHOULD drop within 36,321 runs, WILL drop.

Also, this is not true. You could implement the method I used in your previous thread instead of the odds. This would give you the number of runs in which you can "expect" to get an item.

Then if you did your fitting on those figures you would come up with an equation with a similar slope but a lower intercept (I'm guessing here)

You would end up with a slightly lower number of runs to get your 40 items.

 

[Bugzy]

interesting

I didn't quite understand the relevance of the didkin system though



Also, this is not true. You could implement the method I used in your previous thread instead of the odds. This would give you the number of runs in which you can "expect" to get an item.

Then if you did your fitting on those figures you would come up with an equation with a similar slope but a lower intercept (I'm guessing here)

You would end up with a slightly lower number of runs to get your 40 items.

I think he is going along the lines of 'i know it will never happen' but there is a chance of it happening.

 

[GooberGrape1]

I used ATMA's probability calculator to figure the "odds" of an item dropping. If I'm correctly informed, ATMA actually calculates the 50% breakpoint, beyond which the probability of the item dropping is greater than the probability of the item NOT dropping. Rather than use the Gotterfunken system (is that right?) and calculate every drop on my own, I just went with ATMA's numbers. Good enough for me.

By assuming that Meph would only drop one item at a time, yes, it increased the intercept while the slope remained the same. I used this method, because although there is a slight chance of dropping more than one item at a time, there is also a slight chance of a particular item dropping several times, while other items just won't drop at all. Again, good enough for me.

The Dynkin system (I hope that's the right name) was used simply to portray this: any item that CAN drop, within a set number of runs, WILL drop. I know the pseudo-random number generator can be evil, or it can drop three Stormlashes at once. I know that. But again, good enough for me.

I believe my math is correct. If anyone can prove me wrong, I will be the first to admit it! The right answer is the important thing, and I believe that everyone needs solid numbers so we don't needlessly waste our time or give up too soon.


GG

 

[Gotterfunken]

The "odds" is the ratio of favorable outcomes to unfavorable ones. So if the probability of an item dropping is p, then the odds of that item dropping are p/(1-p). The number that you see in the output is 11-p)/p which is the same ratio. Since our probabilities are so low that for just about any item, if you kill the boss (1-p)/p times you have about a 1/e ~= .368 chance of not getting that item. This means that your estimate of 1000 runs to get the first 40 items is a little low.

There are 40 items that Dynkin system would predict would drop, but I think a more reasonable estimate would predict you only get about 30 of those items in 1000 runs (.632*40 + a few because some have a lot better odds than 1:1000). But if you increase the number of runs to 2000 you have about a 87% chance of getting all the 1:1000 items (first 40) and 63% chance of getting the 1:2000 items (7 more). This will probably give you about 40/47 items.

I just started guessing at the end, but I think the probabilities work out right for at least the first 40 items and 1000 runs. Of course the trend is the next 1000 runs you do will net you a few new items. So it does seem pointless to keep running Mephisto indefinately. The only benifit is how easy and quick the runs are.

I was going to do some better calculations, but I didn't have any batteries for my graphing calculator. I just found Excel on this computer so I might run some numbers in a minute.

 

[sunbearie]

I must say I don't understand the numbers here at all. Might I just ask, GG if you have any tips on easy ways to count how many Meph runs you do? :lol:

 

[Gotterfunken]

Okay, I ran some numbers in Excel. I think these are some reasonable assumptions that we have been using.
1) ATMA drop probabilities are correct (with players 3 and 500 mf)
2) Each kill is a mutually exclusive event (we can just multiply everything)
3) Every item could possibly drop at any kill (Too hard to work any other way)

Of course 3 isn't totally accurate, because we could at most get 6 of our items in one kill. As you have shown, it is extremely unlikely that even 2 of them show up in the same drop. I don't think this asumption will extend the optimal number of runs any. But just think, if you are getting an elite unique every run... are you going to stop running anytime soon? Even if they are duplicates?

The numbers:

runs      1k       2k       3k       4k       5k
avg %     63.24    78.77    84.78    87.95    89.99
EUs       37.31    46.47    50.02    51.89    53.10
% > 80    22       43       45       48       50

Analysis:
I calculated the percent chance you would see each elite unique at least once if you killed Mephisto thousands of times under our assumptions. The first line shows the average chance for all of the 59 EUs. The second line is the expected number of different EUs. The third line is the number of EUs that have a % drop chance greater than 80%.

So the general trend is as expected and increases as more kills are scored, but at a decreasing rate. For the first 1k runs, the numbers are between my previous estimate and GG's number. I also included a number similar to shermo's reasoning, but I increased the needed % from 50 to 80. It seems to easy to be unlucky and not score the EUs if the expectancy is only 50%.

If you make 1k runs, you'll get nearly 2/3 of the EUs that Mephisto can drop. But if you are unlucky you could get considerably less. Adding another 1k runs will net you only about 10 more elite uniques, but it will increase the chance that you get some of the middle items quite a bit. No matter how many times you kill, the chance of dropping the rarer items still remains low (10% chance for Stormlash after 4k runs) .

Conclusion:
GG's estimate of 1000 runs is pretty good. You'll get most of the uniques, but most likely not see anything very rare. If you want to get about 10 more uniques, do 1000 more runs. If you desperately need something that Mephisto only rarely drops, then run him. He's also a good source of upgradable rares (I just found a 429ED/20IAS partizan. I'm not sure if that's worth anything). If you've found all but the last dozen rarest items from Mephisto, it might be more profitable to move on to the next big target to open up some better drops.

 

[Gotterfunken]

I must say I don't understand the numbers here at all. Might I just ask, GG if you have any tips on easy ways to count how many Meph runs you do? :lol:

You could take a screenshot after every run.

Or you could take a shot after every run and measure the amount of alcohol you drank the night before and divide by the shot volume. This might reduce the amount of overall runs you can do.

 

[GooberGrape1]

No matter how many times you kill, the chance of dropping the rarer items still remains low (10% chance for Stormlash after 4k runs)

I agree with you, Gotter. Meph isn't worth running for the very rarest items for two reasons:

1) he simply can't drop TC87, so you WONT get the best items, no matter how many runs.
2) he's a very good source of NON-rare items, which defends the less = more philosophy.


GG

 

[shermo]

I must say I don't understand the numbers here at all. Might I just ask, GG if you have any tips on easy ways to count how many Meph runs you do? :lol:

work out how much xp you get for killing mephisto once. Then record your total xp gained in a sess and voila

 

[Silverbolt]

I must say I don't understand the numbers here at all. Might I just ask, GG if you have any tips on easy ways to count how many Meph runs you do? :lol:

I haven't done anything close to this scale, but I used a simple method for keeping up with run numbers when I did a Mephisto Runs giveaway. The char I was running with used a rare ammy with Teleport charges. I had a map that only needed 1 charge per run (across the moat), so I had a built in counter to make sure that each person in the giveaway got their ten runs worth of stuff. I just recharged the ammy after every 10th run for simplicity's sake, but you wouldn't have to do that yourself.

--SB