The probability that I got bored is 100%

Thank you to VT2WA for your contribution, and even if it's not up to snuff for a few people, this does show exactly what happens to me almost every single game. It'll either be something like 8 lands in top 12, or 4 lands in top 20. Until this games shuffling algorithm is fixed, it will remain a complete joke to me. If the devs cared at all about integrity, this would have been addressed a long time ago.

Edit: Let's also not forget that this game has a free shuffle, which to me, indicates that they might be aware of some kind of issue. Also, in the demo, they clearly had control over the land distribution as you would never get flooded or deprived in opening hand.

So, if this happens "almost every single game" and for me it happens once every 20-30 games, I think we have an explanation!

The game hates you. Sorry.

How did you shuffle the real cards ?

Correct procedure to approach randomization is 7/8 x riffle shuffling.
The 400 sample is low it would be helpful to run this through a hypergeometric sim to make sure though.

These are the results of a 1692 games run in the simulator, for 17 drawsteps each game, with 26 land.
The datapoints represent the probability to have drawn X amount of lands by drawstep 17.


X Dist Data
0 0.00 0.00
1 0.00 0.00
2 0.00 0.00
3 0.01 0.01
4 0.04 0.04
5 0.09 0.09
6 0.17 0.17
7 0.22 0.23
8 0.21 0.20
9 0.15 0.14
10 0.07 0.07
11 0.03 0.03
12 0.01 0.01
13 0.00 0.00
14 0.00 0.00
15 0.00 0.00
16 0.00 0.00
17 0.00 0.00
Mean 7.37 7.34
SD 1.74 1.77



blue surface represents the distribution, red surface the result of the sim at 1692 runs. As you can see, your real life results approach this pretty well, but your sample size makes the curve less smooth.

Beyond that, your findings don't need deep analysis to be interpreted : Duels does not appear to follow a hyper-geometric distribution. Both screw (<30% of lands drawn) and extreme flood (>70% of lands drawn) seem to be dis-proportionally common. Basically the game is just borked, and always has been.

This is exactly why fake news works. Here's some garbled results, which probably mean nothing, and certainly haven't been handled properly (at least in terms of presentation). A couple folks challenge the results (with very good justification btw), even offer to fix them... the rest of you... Let's build a wall!!!!!

alright DJ. what's the problem ? You gonna tell me why it is all wrong or you just going to post vague crap like this. At least I offered to add to the experiment by running the scenario through a sim... whilst you were still deciding what distribution it is supposed to be following. Given you were looking for a model that describes probability of k successes in n draws, without replacement, from a finite population of size N that contains K successes, to my casual mind the choice is rather straightforward.

Personally only flaws that stood out to me from the OP were the garbled presentation (specifically, he does not mention what is his value for n) and the low sample size of manual games. There is obviously no guarantee that his measured data from 1700 games has not been incorrectly recorded/ falsified; but unless you are willing to record the same sample size, there's no way to either confirm or deny that

You? No. But I told the OP, who has some math background, exactly how he screwed things up. As I said yesterday, his data mixes different n's and weights them equally (probability of drawing 0 lands for 30 turns is not the same as the probability of drawing 0 lands for 3 turns. You know exactly how to test this statement, and you also know it's true.). Furthermore his graphs mix probability bands, which should not be mixed: 30-40 and 40-50% are NOT equal likelyhood bands when the mean 43%... this is a little trickier, but I tried to explain it layman' terms.

I even offered to help him out AND test the data appropriately for him. This btw is a trivial exercise to do, and might prove definitively that there is a problem with the shuffler, or it might not. But either way, right now we just have a garbled mess. A mess I can fix, which is why I offered.

@the second part of your post, assuming he kept a record of n for each data point, which his OP implies, I CAN fix this, and even report an exact probability that the shuffler is screwed up. It's just a matter of either him doing it based on what I explained to him, or sending his data my way.

I love it when you talk dirty

I'm sorry DJ, I missed that post on page 1. The banding did not occur to me, I just caught the lack of n within his presentation, which you also cover.

I do think that since he is not representing a number of draws, but rather a percentage of lands drawn OVER THE ENTIRE EXERCISE, that his results, when graphed, should resemble the bell curve we get from the sim I displayed above. Which checks out for the real games (although his percentages are different, this may be simply because of his games running for 5/6 turns instead of my proposed 10, his results seem to be quite consistent with the simulated plot), but does not for the Duels sample, although the banding might have skewed the form of the graph quite a bit.


Maybe the sim i shouldve ran was expecting the games he played with 'real' decks to last 6 turns :

(1700 runs)
X Dist Data
0 0.00 0.00
1 0.00 0.00
2 0.02 0.02
3 0.07 0.06
4 0.15 0.16
5 0.23 0.23
6 0.24 0.23
7 0.17 0.18
8 0.08 0.09
9 0.03 0.02
10 0.01 0.01
11 0.00 0.00
12 0.00 0.00
13 0.00 0.00
Mean 5.63 5.65
SD 1.59 1.59

this again seems to be not a good fit however. So yeah.... maybe you are right and the data isn't usable if he does not define the n variable.

The thing is it's not a bell curve - even in your graph. And the center of the (not actually a bell) curve depends on n (which you can plainly see on the site you linked to - try n = 11, 12, 13, and 14 for example). Same thing happens, btw, if you group them by percentage of land drawn - again the results and their meanings depend on n - For example if I drew my entire deck probability land drawn = 26 is 100% (I have a 100% chance to draw 100% of my land), if I drew only 11 cards, that probability is much much lower (almost 0). But the graphs put all of these apples, oranges, and tomatoes together.

It is in effect 0 cant have 26 potatoes with 11 oranges ! MATH !

hah! You say potato, I say false equivalence.

Some clarifications on the method of data collecting for Duels:
- Again, commit before the game starts that I keep opening hand no matter what and play with it (no matter how painful: like an Esper draw-go control deck, that goes for 14+ turns of me drawing land).
- I keep track of all the cards coming off the top. I can scry something to the bottom, but I still keep track of the Land - Non-Land count coming off the top.
- I track the number of cards I draw off the top, and whether they are Lands or Non-Lands.
- I didn't state this before: but if I played a mill deck, it wouldn't count regardless of win or loss. The games I lose against a mill would skew the results since obviously at the end of the game would be a perfect 26 lands out of 60 cards.
________________________________________________________________________________

Some clarifications on the method for real life data:
- paper cards, 26 lands, 34 non-lands.
- Shuffle for at least a minute.
- I grabbed a bunch of the papers I was using to track Duels data. I then would "copy" the number of cards drawn in a game. Example: if the first Duels game on the data paper I had was 20 cards drawn. After shuffling my deck, I would flip the first 20 cards off the top, and I would count Lands vs Non-Lands.
- After record results. Shuffle the deck for at least a minute.
- Shuffle method included swirling all the cards on the floor (whatever you call it), the repeated rifts and hand over hand, until the minute elapsed.
- I understand n = 400 isn't perfect. But you are welcome to add to the data set by sitting on a floor listening to a movie, or sitting at a table listening to TV, for 6+ hours to add an additional 400 data sets. (Again, wife was gone and I had a free day, doesn't happen often)

_____________________________________________________________________________________

Trying to take in to account some of the comments:

- New graph with % distributions. Keep in mind the % of the bars have to be equal, or else the data WOULD be skewed. If I tried 0 - 20, 20 - 35, 35 - 50, 50 - 65, 65 - 80, 80 - 100, or something similar, some of the data sets have a larger representation and the graph would be skewed.

****I will attach the raw data in a minute***** (assuming I can attach an excel file?)

The following is the graphs redone and normalized with greater increments of percentages (smoother curve)

I tried to attach the raw data, but it won't allow .xlsx files. Message me an email and I'll send it to you.

(I'm a structures engineer guy, don't even try to go through some coding method or some complex online process for posting, I'll just email it to you)

Pm sent.

Raw data sent.

Okay, so some things that come off immediately. We can reject with 99% confidence that the true mean of the population is .43 at basically every level of n except a couple. That's an odd finding, since we know that the true mean is .43. This isn't an indication of a broken shuffler with fat tails, this is an indication that the shuffler consistently prefers to put land toward the top of the deck - in other words a 26 land deck is closer in terms of its performance to a 30 land deck.

That's not just a problematic shuffler, that's a complete systematic failure.

I'm not trying to start trouble here, but are you sure the deck had 26 lands in it? And no ways to consistently draw additional lands. Because if not, the duels shuffler doesn't work at all. And I don't mean slightly broken, I mean completely and obviously broken.

If you hadn't told us that the deck had 26 lands in it I would reject that possibility with near certainty. The minimum number of lands I'd even believe (had you not explicitly told me otherwise) is 28, but the data indicates 30.

Mean:.4956765
SError:.0124345
99% CI: (.4619118 .5294411)

FYI, I clustered the data by cards drawn, as it was supposed to be, in case you want to find how I got the exact numbers - doing so increases the Confidence Interval, so if we don't think it's necessary, then the rejection is even stronger.

For the record, these are shocking results.

Let's be a little more clear here: this absolutely suggests that you would always expect to draw too much land. It's not suggesting that low lands lead to few land draws, or high lands lead to high land draws - it's suggesting that no matter what you'll draw too many lands. It damn near suggests that mana screw (not flood) is extremely unlikely - should basically never happen.

Can you generate a few plots for the most common n's to visualize the duels behavior a bit better (i'm assuming in the raw data he sent you, he noted the amount of turns played per sample) ?