November 10, 2019 November 14, 2019
The Pirate101 Morphing System
(Don’t) Tell Me The Odds
If you’ve played Pirate101 for an extended period of time, you’ll notice that players with “perfect pets” are few and far between. On the other hand, in the Hatchmaking Kiosk alone, a Wizard101 player will notice a plethora of perfect pets. Is there a legitimate discrepancy between the difficulty of obtaining a perfect pet in each game? Or can one blame this difference on a dearth of Pirate101 players? In this article, I’ll do a statistical comparison of the morphing/hatching process in these games in order to answer this question. Afterwards, I’ll discuss some potential changes to the Pirate101 system to help ameliorate the difficulty in morphing.
A Few Assumptions
Before I get started, there are a few assumptions that I need to make about creating pets in both games in order to make the math possible. While they simplify the system to an extent, I don’t think that they will ultimately affect my investigation’s results.
- A pet is equally likely to inherit every talent in its pool, regardless of its overall pedigree. This assumption does away with the notion of “sticky talents” as it otherwise makes any concrete calculations impossible. While sticky talents definitely exist, there are good sticky talents and bad sticky talents. I’m going to assume that these essentially cancel each other out.
- A Pirate101 player going after a perfect pet wants a specific set of 4 talents and 3 powers.
- Wizard101 players going after a perfect pet want either 5 normal talents or 5 derby talents — they’re not trying to perfect both sides of the same pet.
Before I get started discussing the two systems, note that the rest of this article will feature lots of math, particularly combinations and hypergeometric probability. If you want an overview on that to better understand this article, read the beginning of this piece.
Pirate101 Morphing
We want to find the probability of morphing into a pet with the perfect 4 talents and 3 powers. Just as a reminder, a Pirate101 pet’s pool consists of 10 talents and 10 powers. It learns 4 of these talents and 3 of these powers by max level. At the start, I’m going to assume that there are exactly 4 talents and 3 powers that you’re satisfied with. Later on, I’ll talk about what happens when there are, say, 5 talents and 4 powers that you’re happy with. Like with any probability question, our answer’s basic skeleton is:
Number of Successes/Total number of Outcomes
Number of Successes is easy. We want 4 specific talents and 3 specific powers. Without doing any math, it should be 
clear that we have exactly 1 success. We can verify this by calculating 4c4*3c3=1 (combination notation, read the piece I linked above if you need a refresher).
Total Number of Outcomes is a bit trickier. An initial reaction may be to calculate 20c7 — after all, there are 20 talents and powers together and 7 will be inherited. However, this figure allows for a pet to inherit 7 powers and 0 talents and other impossible combinations.
Instead, we should think about it as inheriting 4 of 10 talents AND 3 of 10 powers. Since talents and powers are independent events (the outcome of one does not affect the other), we can multiply 10c4 (10 talents, choose 4 of them) by 10c3 (10 powers, choose 4 of them). This yields a monstrous total of 25,200. A perfect pet is exactly 1 of those outcomes, an approximately .004% chance. To put this in perspective, hatching a pet in Wizard101 has a mere 252 outcomes by the time it reaches mega.
Increasing Your Flexibility
So what if there are, say, 5 talents and 4 powers that you’re ok with having on your perfect pet? Does this increased flexibility make a major difference? Also, how flexible would we have to get to have a .5% chance of a perfect pet? A 1% chance?
Let’s start with 5 talents and 4 powers. Our total number of outcomes is the same, 25,200. Now, we have 5c4 talents on any perfect pet AND 4c3 powers on this pet. Multiplying those together and dividing yields a result of 20/25,200, around a .08% chance. To get a perfect pet by morphing, you should expect to have around 1250 attempts.
How much more flexible would we need to get to have a 1% or .5% chance of getting a perfect pet? This would take 100 or 200 attempts on average to get favorable results, respectively. For the 1% case, the numerator of our fraction needs to be at least 252. One such solution would be to have 6 talents and 6 powers that you’re ok with. (6c4*6c3=300) For .5%, the numerator needs to be at least 126. An example solution is having 6 talents and 5 powers that you’re satisfied with. (6c4*6c3=150)
All Your Perfect Imperfections
What about imperfections? Maybe there are only 3 talents and 2 powers that you really need. You’re fine with having 1 “fail” talent and power. As always, there are 25,200 total outcomes. The numerator is a bit trickier. It’s essentially a case of hypergeometric probability. We have 3 good talents that we need (3c3) and 2 good powers that we need (2c2). The last talent comes from a pool of 7 (7c1) and the last power comes from a pool of 8 (8c1).
I’m going to assume this pool has both good and bad talents in it, but that you’re essentially indifferent to which ones you get. It’s certainly possible to separate the good and bad talents and calculate the probability of getting AT LEAST 3 good talents and 2 good powers, but that makes the calculation needlessly tedious.
So, after multiplying, our fraction looks like this ((7c1*3c3)*(2c2*8c1))/(10c4*10c3). If you look closely, the expression is two hypergeometric probabilities multiplied together (since talents and powers are independent). This yields a result of .2%. Thus, if you only need 3 talents and 2 powers and assume away stickiness, you’ll take an average of 500 morphing attempts to get a pet you’re happy with.
A Word of Caution
These numbers underscore the importance of having a strong pool of talents and powers when morphing. If you have more satisfactory talents in the pool at large, you’re much more likely to achieve desirable results. However, you can’t always afford to be flexible. For instance, a buccaneer might only want Relentless, Elusive, Turn the Tide, and Riposte. Two of these 4 talents are complete necessities on any melee pet (relent and elusive) and this buccaneer may find the additional options (First Strike, Blade Storm, Witch Hunter) not worth their time.
Furthermore, I think it’s worth asking: has the assumption of no sticky talents vastly altered our solution? Having even one good sticky talent and power in a morphing project increases your odds by nearly 10 times (proving this is left to the reader, should they be interested)! However, I think my overall point still stands. Even in a best case scenario of 1 sticky power and talent, there are still over 3,000 potential outcomes. As we’ll soon see, this is far more than in the Wizard101 system.
Wizard101 Morphing
In Wizard101, a perfect pet is one that has 5 specific talents (either normal or derby, the side we focus on is irrelevant for the calculations), out of a total pool of 10 talents. For now, I’m going to ignore jewels, but I’ll factor them in later. Once again, I’m going to assume that there are exactly 5 talents that you’re happy with (this assumption can be changed later, using the same logic as in the Pirate101 case). So once again, our formula is:
Number of Successes/Total number of Outcomes
We’re choosing 5 talents from a pool of 10, so the total outcomes is 10c5. Since we want 5 specific talents, we only have 1 successful outcome. Hence, the odds are 1/252, or around .4%. It’s easy to see that even in the most basic case, assuming away pet jewels or additional flexibility, Wizard101’s hatching system yields 100x better odds than Pirate101’s morphing system. Recall that even if we were ok with 1 talent and 1 power FAILING in Pirate101, we only had a .2% chance of success. I’m completely shocked that these two systems have that much of a disparity in difficulty.
Jewels!
As you probably know, a 2015 Wizard101 update introduced Jewel Collars onto pets. These jewel collars allow wizards to add 1 talent onto each of their pets. How does this change the probability of a perfect pet? First, I’m going to assume that any talent you can add via jewel can also be found in your pet’s pool and vice versa. This means that there are 6 talents that you need on your pet, and you need to choose any 5 of them to manifest on your pet. The 6th talent will come from the jewel. Thus, the total successes is written as 6c5. We have the same number of total outcomes, 252. Our total probability is 6/252, or about 2.3%. Note that if the talent you’re “jewelling” is exclusive to a jewel (an easy example is one of the Kroger card jewels), the odds are unchanged from the first scenario.
This case of jewels essentially represents the current Wizard101 pet system. Thus, assuming that all 6 of your desired talents are in your pet’s pool, you’re able to jewel the 6th talent, and sticky talents do not exist, you’ll only take around 50 hatches to get a perfect pet. Furthermore, they can be further increased if you have an additional talent that you’re happy with. This would make the numerator 7c5 and the total probability 8.3%.
Fixing the Pirate101 Problem
The Wizard101 system offers great odds, and I don’t think KI should change anything about it. However, it’s clear that the Pirate101 morphing system is significantly more difficult, and I think change is warranted. The main priority should be to offer better chances to Pirate101 morphing enthusiasts, who are currently faced with nearly insurmountable odds. I’ve seen proposals that focus on making pet training quicker. While I’m not opposed to this per se, this doesn’t fix the fundamental issue: A Pirate101 player will, on average, take thousands upon thousands of morphs to get a completely perfect pet, unless they are incredibly flexible. I’d much rather train more slowly and only take hundreds of attempts than quickly and take thousands.
Jewels For Pirates
I think this is the simplest solution, as the framework already exists in Wizard101. Not only that, it makes flavorful sense too- pirates love collecting jewels! Let’s start by calculating the probability of morphing 5 perfect talents and 4 perfect powers, given that you can jewel both a talent and a power. The total outcomes are the same- 25,200. Now for the successes. The logic, and therefore the solution, are the same as the case where we were okay with 5 of our pool’s talents and 4 of our pool’s powers- 1/1250, or .08%. Still pretty terrible.
What about getting 4 perfect talents and 3 perfect powers, given that we can jewel 1 talent and power? There are still 
25,200 outcomes. The number of successes is pretty complex, so bear with me. We need to calculate the odds of missing 1 talent and 1 power, only 1 talent, only 1 power, and a completely perfect pet. Then, we sum these 4 probabilities. Why? We can jewel UP TO 1 talent and power. If we get a perfect 4 talents and/or 3 powers, this is unnecessary.
Let’s start with jeweling 1 talent and 1 power (missing 1 talent and 1 power). First, we have 4 talents that we need, and we need to train 3 of them (4c3). For the last talent, we pick 1 of 6 (6c1). We use a similar logic for the powers. 3 powers needed, we pick 2 (3c2). We pick 1 of 7 remaining powers (7c1). After multiplying these 4 values together, we get 504/25200, or 2%.
Now for jeweling 1 talent and no powers (a perfect 3 powers). Using a similar argument, there are 4c3*6c1 successful combinations of talents and 3c3 successful combinations of powers. Multiplying the values together yields 24/25200. Similarly, jeweling 1 power and no talents has 21/25200 successes (4c4*3c2*7c1=21). We’ve already determined that there is a 1/25200 chance of a perfect pet. Thus, the overall odds of a perfect 4 powers are (504+24+21+1)/25200, or about 2.2%. These are pretty reasonable chances that I’d be satisfied with.
There Can Only be One (Jewel)
What if we can only jewel 1 talent or power? How does this affect the probability of morphing success? Well, this means that the pet in question needs to be missing 1 power and no talents, OR 1 talent and no powers. This is a union of two events, meaning the probability is equal to the sum of their individual probabilities, minus the probability of both events occurring simultaneously. This last probability, called the intersection of the events, is 0. Why? It’s impossible to be missing 1 power and no talents AND 1 talent and no powers at the same time. We already calculated the individual probabilities above, so the odds are 24+21/25200, or about .18%. I don’t think these odds are satisfactory (less than 1/500), so I think allowing 1 talent AND 1 power to be jeweled is necessary.
Rerolling
In addition to jewels, there’s also the option of offering rerolls on one talent and one power on a max level pet. This could be done for free once for talents and once for powers, with subsequent rerolls costing an increasingly large crown fee. For instance, I could opt to reroll a selfish pet power and talent, hoping to transform those into my desired talents. What are my odds of getting a perfect pet in this scenario? First of all, I would need at least 3 of the talents and 2 of the powers to already be perfect. From there, I would need any necessary rerolls to be successful. Like in the previous set of examples, we’re going to be looking a 4 scenarios: missing a talent and a power, missing only 1 talent, missing only one power, and missing neither a talent nor a power (an already perfect pet).
Scenario 1: Rerolling a Talent and a Power
We already calculated the probability of missing a talent and a power in the jeweling section- 504/25200. Now we need to find the probability of a successful reroll. I’ll assume that you wouldn’t be able to reroll into the talent/power you just had or one of the other already-manifested talents. This means that there are 6 outcomes on any given talent reroll and 7 on a power reroll. As always, I’m going to assume that there’s only 1 good talent and power remaining (you’ve manifested the others already). So, your odds are 1/6 for talents and 1/7 for powers. Since the events are independent (one happening doesn’t affect the other), the odds of both happening are 1/42. This means that if you reroll a talent and a power once, your odds of getting a perfect pet are 1/42*504/25200, or 12/25200.
What about fixing 1 of the talents or 1 of the powers? This is a union of two events, so the total probability is equal to the sum of each happening individually, minus the probability of both happening. This is 1/6+1/7-1/42, or 12/42. Thus, if you reroll a talent and power once, the odds fixing at least one of the “fail” talents is 144/25200, around .5%.
However, if you are allowed subsequent rerolls (for a fee), you will eventually get the power and talent you want. Thus, the long term odds are equal to 504/25200. This is because you are GUARANTEED to get the last power and talent if you reroll enough.
Scenarios 2 and 3: Rerolling a Talent or a Power
We already calculated the probability of missing only a talent- 24/25,200. We also calculated the probability of successfully rerolling a talent- 1/6. Thus, the odds of getting a perfect pet, given that you reroll your one missing talent once are 4/25,200. Just like before, if you reroll many times, the long term probability approaches 24/25,200.
Now, let’s look at the other side of the coin, missing only a power. The odds equal 21/25,200, as we already calculated. The odds of rerolling a power successfully equal 1/7. Therefore, the overall odds of a perfect pet after one reroll are 3/25,200. If you reroll many times, the long term probability approaches 21/25,200 (using the same argument as before, EVENTUALLY we will roll the correct power).
Scenario 4: Rerolling Neither
Easiest problem of the day, folks. The pet is already perfect, and we know the odds of that are 1/25,200. So, allowing for one reroll of a talent and power, the total odds of a perfect pet are (12+4+3+1)/25200, or .08%. This is still abysmal, but much better than 1/25200
However, if you reroll enough, those odds approach a very reasonable (504+24+21+1)/25200, or 2.2%. This is ultimately the same probability as the jewel example, meaning that both solutions are equally valid. I would welcome either change to the game with open arms, as it would finally give Pirate101 morphers a reasonable chance of getting their dream pet.
Conclusion
Since the inception of the Advanced Pet system in Pirate101, morphing enthusiasts have suffered under a system that offers them insurmountably poor odds of getting a perfect pet. On the other hand, Kingsisle has (rightly) improved upon the Wizard101 pet system, giving experienced pet makers great odds of success. I believe it’s long past time for the Pirate101 pet system to receive some help. Adding in changes like pet jewels or the ability to reroll “fail” talents would go a long way towards making the pet morphing system (and arguably the game at large) more accessible to veterans and newbies alike.
