Ooh! I love combinatorics :D
I'll try calculating the odds of these things happening for all the posts that provide enough information to do that. if I got anything wrong, just let me know:
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Raboniel
the chance of not getting the right color is 3/4, so for that to happen 3 times, it's 3/4 * 3/4 * 3/4 = 0.42. That's a 42% chance- not as uncommon as it might seem.
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Ais
oof that's. wow. For this I'll assume that the 11th hatchling got the correct colors, so getting the incorrect tertiary 10 times in a row would be (1/2)^10, which is 1/1024, a 0.98% chance of happening!
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Achaius
Again, a 3/4 chance of not getting the right colors, and 7 dragons so far with the wrong colors. (3/4)^7 = 0.13 = 13% chance. So not really uncommon or really common.
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LouckyKoneko
Well, multiplying each chance together, 1/2 * 1/4 * 1/2 * 1/2 = 0.031 = 3.1% is your total odds of the exact right hatch. 1 - 0.031 = 0.969 = 96.9% chance of a wrong hatch. (0.969)^34 = 0.34 = 34% chance of having that number of hatchlings with no match. This really illustrates how multiplying fractions together is Really Not Ideal, which I'll go into more detail about below
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RheoTastic
Similar calculation as some of the ones above, (11/12)^44 = 0.022 = 2.2% chance. Oof.
Now for some general principles of combinatorics and probability that I find useful when doing my own breeding projects:
Say you have a project pair that has a range of 1 color above your goal to 1 color below your goal for primary, secondary, and tertiary. for the exact correct colors, that's (1/3)^3 = 1/27 = 0.037 = 3.7% chance. But for each combination that's one color off, it's a 1 * 1/3 * 1/3 = 1/9 = 0.11 = 11% chance, and there are 3 possible combinations like that, so a 33% chance total. nearly 9 times more likely than an exact match!
The numbers won't be the same for every pair, but it makes sense that 1-offs or general close colors are more likely than exact matches, especially if you have a range between colors in multiple slots (primary and secondary instead of just secondary, for example). This is because there is only 1 combination that's correct, but there can be many more combinations that are
almost correct. That's why it can be really helpful to have at least two unrelated preliminary pairs for each dragon in your breeding project, then make more pairs with their one-offs and near-misses. This is a big part of how I finished a breeding pair project for a color combination in 3 different color ranges with 0 active dragons with the combination (when I started) in only 4 months.
Another part of how I did that was understanding ranges and how they work. A dragon that is "5 off" your goal but has 1/2 odds for the primary, 1/4 for the secondary, and 1/2 for the tertiary is 1/16 = 0.0625 = 6.25% chance of the right colors, whereas a dragon that is also "5 off" but only on the, say, secondary, has a 1/6 = 0.17 = 17% chance of the right colors- much better! Not all #-offs are made equal.
To put it in simpler terms, when you only have 1 color slot that's off, you're adding to the denominator of the fractions, but when there are multiple color slots that are off, you're multiplying the denominator of the fraction, which gets bigger much faster! For a visual representation of this, look at the graphs of y = x versus y = x^2.
But of course, the last part of it was simple RNG, which everyone can see by how much the odds I calculated for each person varied, from 42% to 0.98%! For that, I simply wish everyone luck, haha
it was a hard thing for me to learn bc it goes against like...visual reasoning, that pairs with ranges like 2/2/2 actually have a lower chance of getting the goal than 1/7/1, even if they look closer--thats why I invested in teaching myself this stuff. but sometimes even knowing this much cant save you
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