tl;dr; The probabilities of obtaining at least two pure offspring when breeding two pure Axies twice is 7.97%, three times 19.41%, four times 31.75%, five times 43.52%, six times 54.09% and seven times 63.2%
Update: A new article is available with a more realistic model and probabilities.
One way to make money on Axie Infinity is by breeding Axies and selling the offspring on the marketplace. To have a constant revenue stream I decided to create a breeding farm that followed the ABC method where with the purchase of only three parents you could set up an infinite breeding loop.
To guarantee that the offspring of my farm would have the body parts I was looking for I purchased three 100% pure Axies assuming that any offspring would also be 100% pure and thus more valuable. Buying 100% pure and virgin Axies is quite expensive and I figured that given that I only needed two offspring to preserve the population I could go for Axies with a high breeding count (5 max). I then bought two 100% aqua Axies, one with a breeding count of 4 and another one with 5. The result of breeding them twice is shown below.
Note: For brevity I will use for now on the term “pure Axie” to refer to a 100% pure Axie and not only an Axie with all parts of the same class as it’s usually the meaning of the phrase.
Figure 1. First generation of Axies
To my surprise only one of the offspring was pure while the other inherited an unexpected gene in the R1. That Axie was no longer suitable for the breeding farm as it would decrease the purity of its offspring and the farm in general. I decided to sell that Axie and try again with the other pure offspring.
I went back to the marketplace and got a very expensive pure virgin aqua Axie with similar body parts to breed with my only suitable offspring. This time I was aware that it’s not guaranteed that the offspring of two highly pure Axie would be pure as well but I figured that the probability would be high anyway, after all my previous success rate was 50%. Given that I only needed two offspring to be pure to continue the farming loop I decided to breed the Axies five times assuming that my chances of success were high. The result of the new breeding process is shown below.
Figure 2. Second generation of Axies
Ouch. Only one of the Axies turned out to be pure. Clearly there was something I was missing.
Turns out there is a 10% chance for each recessive gene (R1 or R2) to mutate when passed down to an offspring. 10% doesn’t sound like much but this is for every recessive gene. To understand how much this would affect the outcome of breeding we can create a formula that answers the question, what’s the probability of getting a pure Axie when breeding two pure parents?
It’s only 28.24%! No wonder why I was having such bad luck with my farm. On my first attempt I was actually lucky to get one pure offspring. Initially I was expecting both offspring to be pure but the probability of that happening was very small.
The probability of obtaining two pure offspring by breeding two pure Axies twice is:
Only 7.97%, it was almost impossible. I was so wrong. Surely if I increase the number of times the parents are bred my chances of getting two pure Axies would also increase, but by how much?
Knowing the probability of getting a certain number of successes (an offspring to be pure) in a certain number of attempts (breeding times) is modeled by a binomial distribution formula shown below.
Using the formula, if we want to calculate the probability of getting two pure Axies after breeding the parents 4 times we can do:
The result above tells us the probability of getting exactly two pure offspring on 4 attempts but what we really care about is the probability of getting at least two pure offspring. To do so we need to tweak our formula as follows:
We are ready now to calculate the probability of getting at least 2 pure offspring when breeding 4 times:
With this formula we can summarize the probability of getting at least two pure offspring to continue the ABC breeding loop when breeding the parents 2 to 7 times.
| Breeding times | Probability of getting at least 2 pure offspring |
| 2 | 7.97% |
| 3 | 19.41% |
| 4 | 31.75% |
| 5 | 43.52% |
| 6 | 54.09% |
| 7 | 63.2% |
If you are curious, the general formula to calculate the probability of getting at least k pure offspring by breeding pure parents n times is:
Expanding the formula we get:
Conclusion
Maintaining a breeding farm of pure Axies is a very difficult task even if starting with pure parents. The 10% probability of each recessive gene to mutate during breeding rapidly decreases your chances of getting two pure offspring to maintain the farm. Breeding two Axies seven times comes at a great financial cost for only a 63.2% chance of success.
The formula and probabilities obtained in the article are good approximations but not the real numbers. For simplicity we didn’t take into account that when a gene mutates it could very well mutate into a body part of the same class. This technically increases our chances of success although only slightly. I don’t expect more than 5 percentage points of deviation from the results obtained here.
Note: The difference is in fact higher, close to 10 percentage points. The updated model can be found here.
Finally, pure breeding farms are falling out of favour given not only it’s difficulty but also the rise of meta builds with interesting combinations of body parts and cards from different classes that in turn change from season to season.
Special thanks to my dad José Barreto for helping me with the math on this article. José is a mathematician with a Master’s degree from the University of Texas and the author of the book “Algebra Lineal en Contexto” (Linear Algebra in Context) that works as an online private tutor. If you are looking for help in math during College or need help preparing for getting into one get in touch with him, he’ll be happy to help you and he has very fair rates. He teaches in Spanish and English.
Aún que no puedo entender los detalles me complace ver aplicaciones de la distribución binomial