tl;dr; Using a more realistic probabilistic model that takes into account destructive and neutral mutations, the probability of obtaining at least two 100% pure Aqua offspring from two 100% Aqua parents are 15.21% when bred 2 times, 34.24% when bred 3 times, 51.32% when bred 4 times, 65.14% when bred 5 times, 75.61% when bred 6 times and 83.24% when bred 7 times.
In the last article we came to the conclusion that the formula to model the probability of getting at least k 100% pure offspring out of 100% pure parents when bred n times is:
Where pk (n,i) is the probability of obtaining exactly n offsprings 100% pure when breeding the parents i times which in turn is modeled by the formula:
Where ppa is the probability of breeding a single 100% pure offspring out of pure parents, defined by the formula:
Where pgm is the probability of a gene mutation (0.1 or 10%) and r is the total number of recessive genes in an Axie (12).
The article closed noting that the formulas and numbers obtained were good approximations to the problem but not exactly correct. The reason is that a gene mutation is described as randomly picking a body part for that gene. This means that even if a mutation happens, the selected body part could be of the same class as before thus not affecting the purity. This gives us two types of recessive gene mutations: destructive (pdgm) and neutral (pngm)
To better understand what that means, let’s look at the number of body parts per class shown below.
| Aqua | Beast | Bird | Bug | Total | |
| Eyes | 6 | 6 | 5 | 5 | 22 |
| Ears | 7 | 8 | 8 | 8 | 31 |
| Mouth | 6 | 5 | 5 | 5 | 21 |
| Horns | 8 | 9 | 8 | 8 | 33 |
| Back | 7 | 8 | 8 | 8 | 31 |
Table 1. Number of body parts per class.
If we are looking for a pure aqua Axie and we randomly pick one body part for its eyes, there’s a 6/22 chance (27.27%) that we pick a body part of the same class (aqua). Alternatively, there’s a (22-6)/22 chance (72.73%) that we end up picking a body part of a different class. The former is a positive as it doesn’t ruin its purity while the latter is a negative as it ends up mixing classes thus decreasing purity.
The table below shows the probability of randomly picking a body part of a different class for all classes and body parts.
| Aqua | Beast | Bird | Bug | |
| Eyes | 72.73% | 72.73% | 77.27% | 77.27% |
| Ears | 77.42% | 74.19% | 74.19% | 74.19% |
| Mouth | 71.43% | 76.19% | 76.19% | 76.19% |
| Horns | 75.76% | 72.73% | 75.76% | 75.76% |
| Back | 77.42% | 74.19% | 74.19% | 74.19% |
| Tail | 74.19% | 77.42% | 74.19% | 74.19% |
Table 2. Probability of selecting a body part of a different class randomly.
Let’s focus again on the eyes of an aqua Axie for a moment. We know that the probability for a gene mutation on any of it’s recessive genes defined as pgm is 10%. Let’s call the probability of randomly picking an eye body part that’s of a different class (not aqua) as pdcy (dcy stands for different class eyes).
pdgmy stands for probability of a destructive gene mutation for the eyes. We can now build a new table that shows the probability of destructive gene mutations for all classes and body parts as shown below.
| Aqua | Beast | Bird | Bug | |
| Eyes | 7.27% | 7.27% | 7.73% | 7.73% |
| Ears | 7.74% | 7.42% | 7.42% | 7.42% |
| Mouth | 7.14% | 7.62% | 7.62% | 7.62% |
| Horns | 7.58% | 7.27% | 7.58% | 7.58% |
| Back | 7.74% | 7.42% | 7.42% | 7.42% |
| Tail | 7.42% | 7.74% | 7.42% | 7.42% |
Table 3. Probability of a destructive gene mutation.
The probability of an offspring to be a pure Axie, (i.e not having single mutation on any of its recessive genes) can be calculated as:
If we keep our focus on an aqua Axie and replace the variables with the values from table 3 we get:
In the previous article where we didn’t consider the effect of destructive and neutral mutations, the probability of breeding a pure Axie was only 28.24% compared to 39.33% with our new model. Using this new value for ppa we can use our formula to calculate the chance of breeding at least 2 pure Axies (k=2) on n attempts. The results are shown in the table below.
| Breeding times | Probability of getting at least two pure offspring |
| 2 | 15.21% |
| 3 | 34.24% |
| 4 | 51.32% |
| 5 | 65.14% |
| 6 | 75.61% |
| 7 | 83.24% |
Table 4. Probability of obtaining at least 2 pure offspring by breeding count.
These probabilities are much better than before and tells us that breeding two parents at least five times gives us a decent chance of preserving the purity of the breeding farm.
Conclusion
I was surprised to see how much of an impact the inclusion of destructive and neutral mutations had on the probabilities of obtaining at least 2 pure offspring. In my previous and simpler model, the probability of obtaining at least 2 pure offspring after breeding two pure Axies seven times was only 63.2% when in reality the probability is much higher at 83.24% using a more realistic model.
No wonder why so frequently I see people breeding Axies at least 4 times in the marketplace, that’s the point where the probabilities of success of keeping the breeding farm pure rise above 50% while keeping the breeding costs as low as possible. I’m also realizing that I was somehow unlucky on my last attempt of breeding two 100% pure Axies five times and only obtaining one 100% pure offspring. Maybe I just need to breed them two more times and trust that I won’t be as unlucky this time around.
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.
Je Je. A la larga la probabilidad se impone…Supongo
Super interesting analysis, David. (You’ve got me pining for my 101 Applied Maths class of many moons ago…) A couple of thoughts of areas to explore. Firstly, getting the same class traits doesn’t mean you’re getting useful traits. Second, often times, a trait from a different class improves the battle-readiness of the axie – as you mentioned in Part 1. Finally, getting a trait from a different class on the eyes and ears is of much less significance.
Yeah I agree wit you. It’s more important to focus on particular traits than in class purity so the actual model would be different to the one I’m presenting here
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