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I’ve seen lots of threads asking about the chances to get certain kinds of cards from packs and optimal extracting / combining strategies. Since one of the ways I enjoy playing games is by picking apart the mechanics, I figured I would share what I’ve found – I hope this helps some folks out.
I’m using {x} to indicate an x-star card.
**PACK CHANCES:**
I’m VIP 0, so I’m only going to be reporting chances for non-VIP packs.
The first section is a model which reasonably predicts the distribution. Certain assumptions were made in the creation of this model. First, it is assumed that the underlying programming uses “round number” probabilities, like 20% or 1/3, and so the modeled values are rounded to the nearest such number. Secondly, it is assumed that the cards are generated using a simple algorithm in which the first 5, lower value cards are generated using identical probabilities and the last, potentially high value card is generated using a different probability.
The second section is a more accurate but less intuitive way to describe the model. These elegant descriptions courtesy of zz1000zz.
The final section includes some values calculated from the raw data. Expected cards/pack is the observed expected number of each card value in the pack on average. A 95% confidence interval is presented, calculated using the normal approximation where the distribution is observed (thus far) to be binomial – note that the normal approximation gives higher error further away from 50%. RoI is the return on investment: the amount of card value (calculated using combine rates with a 45% bonus as in the combine section) you get per silver piece when buying that pack.
**Novice** (Very High {1}; Very High {2}; Normal {3}):
Statistics: 88 packs. One pack contained {4} card, was rejected as outlier.
Rounded Model:
Cards 1-5: {1}(80%); {2}(20%)
Card 6: {2}(80%); {3}(20%)
Accurate Model:
5a = 4.03
5b + c = 1.81
d = 0.18
a + b = 1
c + d = 1
5a + 5b + c + d = 6
a = chance of {1} in slots 1-5
b = chance of {2} in slots 1-5
c = chance of {2} in slot 6
d = chance of {3} in slot 6
Expected cards/pack: {1}(4.03); {2}(1.81); {3}(0.18 +/- 0.08)
RoI: 1.44
**Standard** (Very High {2}; High {3}; Normal {4}):
Statistics: 197 packs. All race packs have similar statistics and are combined.
Rounded Model:
Cards 1-5: {1}(30%); {2}(70%)
Card 6: {3}(80%); {4}(20%)
Accurate Model:
5a = 1.51
b = 0.21
c = 1 – a
d = 1 – b
a = chance of {1} in slots 1-5
b = chance of {4} in slot 6
c = chance of {2} in slots 1-5
d = chance of {3} in slot 6
Expected cards/pack: {1}(1.51); {2}(3.49); {3}(0.79 +/- 0.06); {4}(0.21 +/- 0.06)
RoI: 1.44
Thanks to GETH83 and Ethan, a poster on another board (courtesy of Seizan7), we have more data on the chances to get a {4} from a standard pack. The sample size expands to 695 and we have an expected cards/pack of {4} (0.186 +/- 0.029).
**Elite** (Very High {2}; Very High {3}; High {4}):
Statistics: 51 packs; confidence is therefore lower than for the previous packs. Also note that Elite packs seem to be weighted to provide a subset of Epic cards – I’ve never seen a unique, and certain cards (like Captain, Templar) are more common.
Rounded Model:
Cards 1-5: {2}(50%); {3}(50%)
Card 6: {3}(30%); {4}(70%)
Accurate Model:
5a = 2.75
5b + c = 2.57
d = 0.72
a + b = 1
c + d = 1
5a + 5b + c + d = 6
a = chance of {2} in slots 1-5
b = chance of {3} in slots 1-5
c = chance of {3} in slot 6
d = chance of {4} in slot 6
Expected cards/pack: {1}(0); {2}(2.75); {3}(2.57); {4}(0.72)
RoI: 1.38
**COMBINING CHANCES**
This assumes that the combine chances reported are accurate. I believe they are.
After running the numbers, the expected value for combining cards is equal no matter how many cards above 4 are used as long as you don’t go over 100%. This is because the cost to add another card is equal to 1/3 the base cost of combining, and because on a failed combine one card is left over (so a base combine risks 3 cards). The lower success chances resulting from combining 4 at a time are exactly offset by the chance to gain multiple improved cards. If you just want one copy, your best bet is to wait until you have more cards. If you want multiple copies (such as for more combining), it is better to combine 4 at a time.
Using 100sp as the baseline for the cost of a common, I calculated the amount you could expect to pay (on average) for each of the higher value cards at a range of % bonuses from alchemy and guild, including combine costs. This formated as StarValue(Cost) and uses the average cost using the optimum number of cards combined. I realize that it’s impossible to combine higher level cards with some of these values; they’re just for comparison.
**Bonus 0%:** {1}(100); {2}(460); {3}(3 460); {4}(45 238); {5}(1 343 509); {6}(60 191 743)
**Bonus 12%:** {1}(100); {2}(432); {3}(2 969); {4}(35 405); {5}(950 706); {6}(38 245 195)
**Bonus 30%:** {1}(100); {2}(386); {3}(2 376); {4}(25 247); {5}(596 770); {6}(20 890 095)
**Bonus 45%:** {1}(100); {2}(360); {3}(2 046); {4}(20 021); {5}(432 311); {6}(13 694 993)
**Bonus 65%:** {1}(100); {2}(360); {3}(1 841); {4}(16 262); {5}(315 290); {6}(8 886 824)
**Bonus 100% (Max):** {1}(100); {2}(329); {3}(1 554); {4}(11 856); {5}(197 354); {6}(4 705 052)
**EXTRACTING CHANCES**
I have less data on this section. The short answer is that the most efficient silver to crafting component ratio is to buy novice packs and extract them, but this method does have some inherant randomness. The next most efficient (~90% as efficient) is to buy {1} cards from the reputation store and extract them.
{1} Card
Statistics: 266 cards.
Energy, Soul, Magic: 1(25%); 2(33%); 3(33%); 4(8.4%)
Elixer of Life: 1(9%)
Expected components: 2.26 each of Energy, Soul, Magic; 0.09 Elixer of Life
{2} Card
Statistics: 10 cards
Expected components: 7 each of Energy, Soul, Magic, 0.3 Elixer of life
{3} Card
No data
{4} Card
Statistics: 2 cards
Expected Components: ~125 each of Energy, Soul, Magic. 1 Crafting Book.

The asterisk character is used to mark text as **bold-faced** in Kong Forums. To use ‘`*`’ as an actual asterisk character, you can surround it with the ‘@’ sign.
Otherwise, thank you for the helpful information.

epic in novice should not be simply rejected, you should at least try to find out it’s chance. Even if it only happens 0.05% of the time it’d still increase RoI by 0.01 which by your rounding would be considered as “significant”. While rare, 0.05% is perfectly possible.
Anyway, going off your current numbers, RoI is as follows:
For: 0/ **12** /30/ **45** /65/ **100**
Novice: 1.91/ **1.76** / 1.56/ **1.45** / 1.41/\* **1.29**
Standard: 2.8/ **2.29** / 1.74/ **1.45** / 1.26/\* **1**
Epic: 2.88/ **2.31** / 1.71/ **1.39** / 1.17/ **0.9**
The 45% one is a bit different from yours, but very close. I’m guessing the data you presented in term of expected count of each rarity might be rounded here. But it’s within reasonable range still. As it is, best RoI goes, from low to high rate, Epic, Epic, Standard, Standard, Novice, Novice(granted due to lack of racial cards in Novice, racial standard will be useful forever and ever)

Well, it’s not like I’m going to forget that I got that epic, but the chances are likely to be equivalant to getting a legendary from a standard, which hasn’t happened in 180+ samples. If we can crowd source some statistics we might get a reasonable guess as to the frequency, but as it is the sample size is just way too small.
Yeah, I rounded the data above, and in any case the error is going to be less than the variance in a lot of these stats. Besides the lack of 1.1/1.2 cards, the other thing to consider about novice is that even with the higher RoI, you’re going to be skewed toward cards like squire and fireball which have 1-star versions, while being unlikely to ever get a unique. Of course it’s worth noting that in all these packs, I’ve only recieved two uniques (a Pontif Faol from human race and the elite from the Novice pack was Sharptooth). The distribution is definitely skewed away from the uniques.

> *Originally posted by **[Woader](/forums/302/topics/359275?page=1#posts-7425405):***
>
> Well, it’s not like I’m going to forget that I got that epic, but the chances are likely to be equivalant to getting a legendary from a standard, which hasn’t happened in 180+ samples. If we can crowd source some statistics we might get a reasonable guess as to the frequency, but as it is the sample size is just way too small.
>
> Yeah, I rounded the data above, and in any case the error is going to be less than the variance in a lot of these stats. Besides the lack of 1.1/1.2 cards, the other thing to consider about novice is that even with the higher RoI, you’re going to be skewed toward cards like squire and fireball which have 1-star versions, while being unlikely to ever get a unique. Of course it’s worth noting that in all these packs, I’ve only recieved two uniques (a Pontif Faol from human race and the elite from the Novice pack was Sharptooth). The distribution is definitely skewed away from the uniques.
The thing with epic/leg in novice/standard is that while they are rare, they carry a huge weight, especially for the lower combine boost parts(epic in novice says ~46 RoI on the pack at 0%). It’s nature is similar to lottery jackpot, infrequent as they are, they impact RoI greatly so cannot be ignored. As I was saying, even a 1 in 2,000 chance would affect the RoI by 0.01 on 45%, chances are it’s more frequent than that and can easily carry 0.2 to 0.01 RoI influence depending on combine bonus. Depending on sample size and variance, this might be a significant influence. (specifically, error of above say 0.02 would probably be unacceptable if sample size is over 1000, as variance on the average decreases as sample size increase)
Tbh, I have been buying mostly racial and my experience is that the card with the 1.0 1 stars are actually the hardest ones to get(in epic) aside from unique even if I combine as much as I can. This might be so to offset the novice 1-star-heavy nature. But those were the ones holding me back when I went for 50% collection on sets.
As for uniques being rare, that’s probably partially contributed by the fact that they can’t be combined from lower stuff and you have to get the epics directly. Personally, uniques have been around average in terms of direct epic received.

I appreciate the point about the high value of the epic/legend offsetting the low chance in novice/standard packs as far as RoI. My reason for not including is the high dependance on variance you mention – once I have enough data to decrease the variance down I’ll feel more comfortable including it.
The skewing away from uniques interests me quite a bit. Looking back of the Standard packs, 1 of 38 was unique (~2.5%). In the Elite Packs, 1 of 37 was unique (again, ~2.5%) For an equal distribution based on a quick calculation from the guide page, Standard packs should yield 5/42 (~12%) uniques. For elite packs the expected % would be 17/162 = 10.5%. This makes about my unique rate 20% to 25% of that which is expected, which is statistically significant.

this data is unique in its own way but um do u have yur original notes that you used for this data, that wuld be really helpful for my data collection, if u do culd u reply to me. thx :D

Geth, thanks for the input – it’s good to get further confirmation on the stats.
Nimi, what are you trying to find out? Should I clarify the math above? Providing the raw data is not really feasible, since it’s hundreds of lines of numbers in a spreadsheet (that’s what statistics are for). If you can provide stats on the data you’ve collected as Geth has, it will help to increase our confidence intervals.

> *Originally posted by **[Woader](/forums/302/topics/359275?page=1#posts-7425113):***
>
> If you just want one copy, your best bet is to wait until you have more cards. If you want multiple copies (such as for more combining), it is better to combine 4 at a time.
neat conclusion.
3 common to 1 good, not applicable here;
5 good to 1 rare, applicable, but may not worth the risk for just 1 more copy;
9 rare to 1 epic, applicable, and epic is the main force in any sense and for most players.
so if the conclusion is valid, go combine every 4 rare to increase the count in epic.
19 epic to 1 legend, applicable, huge potential, shoot 4 epic for 1 legend seems very legit.
39 legend to 1 god, gigantic potential, but forget about it, at least for quite a long while….

> *Originally posted by **[juderiverman](/forums/302/topics/359275?page=1#posts-7426188):***
> > *Originally posted by **[Woader](/forums/302/topics/359275?page=1#posts-7425113):***
> >
> > If you just want one copy, your best bet is to wait until you have more cards. If you want multiple copies (such as for more combining), it is better to combine 4 at a time.
>
> neat conclusion.
>
> 3 common to 1 good, not applicable here;
>
> 5 good to 1 rare, applicable, but may not worth the risk for just 1 more copy;
>
> 9 rare to 1 epic, applicable, and epic is the main force in any sense and for most players.
>
> so if the conclusion is valid, go combine every 4 rare to increase the count in epic.
>
> 19 epic to 1 legend, applicable, huge potential, shoot 4 epic for 1 legend seems very legit.
>
> 39 legend to 1 god, gigantic potential, but forget about it, at least for quite a long while….
well, if u are shooting for epic or higher(especially the “higher” part), good to rare in 4s is probably a good idea. Cause you are not just aiming to get 1-2 card for safe use quickly, but instead you are gonna need to make loads of copies in the long run and the odds will even out usually.

The pack distribution stuff is good info.
Your conclusions on card combining EV are wrong, to potentially catastrophic effect for those who take your advice of combining only 4. There is rarely a card advantage to be gained by trying multiple inferior combines instead of one high-chance combine.
![](http://img689.imageshack.us/img689/8826/9myi.png)
As you can see for my current values on rare {3} cards with Lab lvl 12, Research lvl 3, EV declines 13% by attempting two combines at 4 each instead of 1 combine at 8.
If you want to gain an advantage over many combine attempts (enough to avg out the misses) you should be sure not to waste any cards raising your success% over 100%. In my case 8 cards gives me the listed 91.3% and a 9th card would give me theoretically a 113% chance. Since success is capped at 100% I’m actually wasting that extra 13% making my 9th card of lesser value than the other 8. Over time, that lesser value would add up to worse returns for me.
WITH THAT SAID, you are absolutely correct that if you must have 1 of a card you should wait for 100% under all circumstances. Only in the unlikely situation that you happen to end up with 30+ tier {3} cards would I recommend you try to beat the system in the way I described above. It’s probably safe to beat the system on tier {2} cards at my level since they are so plentiful and I don’t care about the outcome. If I combine only 4 instead of 5 there is a small efficiency gain that will eventually add up in my favor. Thanks for the insight on that.

Andrew, your data is not accurate either. I totally agree with you, I use to combine only when I have at least 50%, still you forgot that if I have 4 cards and I combine them 2 by 2, I can do 3 events unless I succed. It’s pretty boring, even if easy, to recalculate everything, since you have to consider when you succed to calculate how many cards you really have left.
Just to make an example, if I have 4 cards I can combine 2 by 2 and succed both, for a chance of 2 successes of 1.7%, or I could fail the first, the second and still have 2 cards for a 3rd attempt, with a total full failure chance of 65.6% (not 75.5%) and a 32.7% chance to succed once (if you succed on the second attempt, you end up with a card left, since you burn only 1 in the first try).
I’d suggest now to calculate again the EV, not considering that you could end up with a spare card.
Well….EV now is 0.361, delta is 3% and, I have to remember this, I did not calculate the chance to end up with a spare card and a combined card (such an occurring should enchance the EV in some way).
If we do correctly the maths, we will see that EV is pretty much the same. Of course, the only REAL difference is that you can fail 100 times in a row with 99% chance and you can succed 100 times in a row with 1% chance. If you feel lucky, go for few cards and many attempts, otherwise follow my path and try to always have a big chance.
But don’t listen to ppl that assume that EV is very different, they are just bad at maths.

Let’s make this simple.
Let’s say it takes exactly 9 copies to get 100%. That means:
2: 12.5%
3: 25%
4: 37.5%
etc
Now, in 4s, if we were to combine 8 times, we’d have:
3 success 5 fails on average, ok?
This consumes 3\*4+5\*3=27 cards, ok?
in 9s, to have 3 success, you need to combine 3 times, that is 3\*9=27, ok?
So to get the same number of cards, the number of material needed is equal, ok?
You need to remember that fails refund a card, ok?
If any of those step seems confusing, please say so.
On another note, the reason why 4 is SUPERIOR to 100% or 100%-1(that is to say the highest copy without overflow) is that it have the biggest portion of cards “consumed” in failure. And when a card is consumed via failure, you get material back. It’s not much, but it builds up and is free. So 4 is, in the long run and assuming you are not hated by lady luck, most beneficial.

I like the way in which you explained with easy words that chance is more or less the same.
Anyway, I would not choose the 4 way only because I get materials worth a bunch of silver…
epic to legendary can award a few books, though, and in that case you got a point.

Shin and deathvonduel have got the right of it. In every case (expected improved cards gained)/(expected inferior cards consumed) is exactly equal; I’ve done sanity checks on some quite complicated instances just to make sure. The denominator is the portion left out of andrew’s calcs.
The conclusions hold: unless you’ve hit 100%, combining less than 4 is inferior because the combine costs do not scale. Combining in more events is slightly superior if you’re combining enough times for the stats to even out because you get crafting components on a failure (basically, instead of just consuming all your cards, some of them wind up being effectively extracted). Combining in fewer events is better if you just want one card because you reduce the variance.

> *Originally posted by **[Woader](/forums/302/topics/359275?page=1#posts-7426742):***
>
> Shin and deathvonduel have got the right of it. In every case (expected improved cards gained)/(expected inferior cards consumed) is exactly equal; I’ve done sanity checks on some quite complicated instances just to make sure. The denominator is the portion left out of andrew’s calcs.
>
> The conclusions hold: unless you’ve hit 100%, combining less than 4 is inferior because the combine costs do not scale. Combining in more events is slightly superior if you’re combining enough times for the stats to even out because you get crafting components on a failure (basically, instead of just consuming all your cards, some of them wind up being effectively extracted). Combining in fewer events is better if you just want one card because you reduce the variance.
I fully agree. To back it up, I did some calculations of expected cost using the following formula:
expected cost = cost of a success + cost of failure \* chance of repeated failures
The cost of a success would be X cards + the combination cost.
The cost of a failure would be the cost of a success, minus the 1 card you receive back.
The chance of repeated failures is the summation, from i=1 to infinity, of failure^i. (I just used 500, since failure^i is approximately 0 by that point, even for Godlike cards).
Graphing this from 1 card to 100%, the graph always plateaus at 4 until you go over 100% (at which point it goes up because of the wasted %). Since the silver/card cost is the same, and you get components back for failures (books when attempting to combine Epics and above), I always use 4s.

Maybe there’s something wrong with my math? The two criteria I use to evaluate card combining is EV of upgraded cards and probability of avoiding total failure (# successes \> 0).
Here’s what that looks like for me personally for the various combination strategies discussed in this thread.
![](http://img546.imageshack.us/img546/723/e1l1.png)
While EV does tend to trend upward as you make smaller and smaller combines you pay for that by lowering your chance of getting any cards at all. There is probably some sweet spot of the two that maximizes EV for the risk of failure you’re willing to take on, but I don’t see why it’s always 4.
If you guys are just optimizing for silver cost and materials, I don’t weigh that in. When I combine cards it’s because I want better cards. If I want materials I’ll dismantle a bunch of 100 silver cards from the reputation shop.

Andrew, I think what you’re doing is only calculating the chances to get one improved card and neglecting the returned card on a failure. If you combine in multiple batches, the decreased chance of getting one improved card is exactly offset by the increased chance of getting 2 or more improved cards.
For example, if you have 8 cards at (for ease of calculation) 10% chance per card.
If you combine all at once, your improved/inferior chances are 1/0(70%) and 0/1(30%). Your expected (improved card gain)/(inferior card used) is 0.7/(8-0.3) = 0.0909 (repeating).
If you fuse in two batches of 4, there are 4 possible outcomes (S=success/F=Fail): S-S(9%); S-F (21%); F-S(21%); F-F (49%). We know this is all permutations since it adds to 100%. This means our improved/inferior chances as above are 2/0 (9%); 1/1 (42%); 0/2 (49%), since we get one card back on each failure. Thus, our expected (improved card gain)/(inferior card used) is (2\*0.09 + 0.42)/[8-(0.42 + 2\*0.49)] = 0.6/(8-1.4) = 0.0909 (repeating).
The final piece is the consideration that below 4, you are paying more per card to combine since the cost is the same from 2-4. Thus the conclusion that if you have enough cards that the statistics even out, combining 4 at a time (to maximize combine events and therefore the component kickback) is the most beneficial. If you only want one of the improved cards, though, combining more at once is better (as you observed), because the pairity relies on the fact that it is possible to get multiple improved cards if you combine multiple times.

Andrew, the calculations I use optimize for the cost of cards and silver used to get a successful combination. I didn’t include the perceived value of materials in my calculations, but used them as a tie-breaker, since everything after 4 cards results in the same EV in cards/silver.
If I ignore the silver cost of combining cards, and just take into account how many cards I may lose in combining, the results I get are still the exact same (plateaus at 4). Also, if you’re combining cards from the reputation shop, then everything basically does become an optimization of silver spent anyway.
I think where our calculations differ is that you count multiple successes and a single success as being of the same value. If you only want to try for a single copy of a card, then I’d recommend using your EV calculations. If you’re going for multiples (especially if you’re making mass quantities of lower rarity cards to eventually combine into a single high rarity card), then I’d recommend using the EV calculations that myself and others have suggested using and always going with 4.
Lastly, in regards to materials: once you start combining Epics, the materials you receive from failures include books, which you cannot get from 100 silver cards. I’d much rather spend the same number of silver/cards on average and get some extra books as a bonus.

My calculations account for returned cards when applicable. That’s why the chance of 1 success when combining 4 at a time has a spike, because if you failed your other 3 rolls you have enough cards left for a 4th attempt.
I think both our calculations are correct but mine operate on a finite number of cards, they show you how to analyze your odds and find the max reward for the number of cards you have on hand. This is a useful analysis for the majority of use cases where you want to improve the cards you are collecting by playing the game.
Your calculations assume a large enough pile of cards that it may as well be infinite, which is good to use when considering how to upgrade reputation shop cards.
My worry is that people will read this and decide that they should make every combine at 4 cards (as one previous poster already has.) This will work for Squires but not for Maia. You have to be comfortable with your chance of total failure when combining Named cards because you won’t have the infinite pool needed for things to average out in your favor.

Andrew, while your calculation is correct, it is only situationally valid due to starting assumptions which are subjective. An arbitrary finite number of cards is not actually the situation with which we’re presented.
The main issue I see with your calculation is even though it uses the returned cards in subsequent combines if there are enough, it does not assign any value to those cards if there are not enough for another combination immediately. I feel that this is accurate only if you just want one improved card – if you want to get more than one, those leftovers will eventually be used when you gain more fodder cards.
My calculation (and the illustrations others have provided) is a statistical analysis, which as you noted shows chances given infinite events. Hence the often repeated caveat that this only is only valid if you’re combining enough cards to be statistically relevant, and that if you only want one improved card, like would often be the case in your Maia example, you should maximize your % chance to decrease variance.
It is worth noting that using more events does not decrease your expected RoI, it just increases the variance. There is exactly a 50% chance that combining 4 cards at a time will use fewer cards than it would take to combine at higher chances (and potentially give you some crafting materials to boot). I tend to prefer more certainty so I wait for more cards if I only want one improved version, but it’s not an objectively bad decision to try your luck.

> *Originally posted by **[YAYitsAndrew](/forums/302/topics/359275?page=1#posts-7427076):***
>
> My calculations account for returned cards when applicable. That’s why the chance of 1 success when combining 4 at a time has a spike, because if you failed your other 3 rolls you have enough cards left for a 4th attempt.
>
> I think both our calculations are correct but mine operate on a finite number of cards, they show you how to analyze your odds and find the max reward for the number of cards you have on hand. This is a useful analysis for the majority of use cases where you want to improve the cards you are collecting by playing the game.
>
> Your calculations assume a large enough pile of cards that it may as well be infinite, which is good to use when considering how to upgrade reputation shop cards.
>
> My worry is that people will read this and decide that they should make every combine at 4 cards (as one previous poster already has.) This will work for Squires but not for Maia. You have to be comfortable with your chance of total failure when combining Named cards because you won’t have the infinite pool needed for things to average out in your favor.
While the number of “higher” cards might be correct, you seem to neglect left overs. 2 at a time have a 87% chance of having a left over lower card(being a “1” it cannot be used again) while 9 and 3 have a 74% chance of having a 1 card left over.
For 8 and 4, the very concept of “8 and 4” is silly. It should be “8 and 4 or 5”
Specifically: if 8 fails, the second combine should be considered as a 5. The chance to not get a card in that case is reduced to 4.1%. Alternatively, in your calculation, should the 5.3% proc happen, you’d have 2 card left over which in themselves can be combined at 13.05% rate, that is to say 5.3%\*13.05%=0.006869, increasing the EV to 1.311.
So your calculation did not take into account the different number of possible left overs as well as the different probability of left over.

In addition, your 2 at a time calculation is wrong. I’m not entirely sure how you did it and how you went wrong, but looking at the “1” case alone is enough to know it is wrong.
Specifically:
There exist a chance that 11 times is tried and all previous failed and 11th is successful:
(1-P)^10xP
There exist a chance that one and only one of the first 10 trials is successful and 11th is NOT attempted because there would be only one card left:
(1-P)^9xPx10 (x10 due to 10 possible order)
where P is 13.05%
This arrives at:
37.07%
and
3.22%
for a sum total of 40.29%
So without going into the more complicated cases(and they are certainly more complicated), this is proof that your calculation is wrong somehow.