I did some calculations on singles versus ten packs, and it turns out packs are still better than singles most of the time. The probabilities of singles versus packs evens out after you have about 3/4 of the legends/myths in the rare egg section.
After you hatch a featured legend in a pack in the 4th spot, the last 6 hatches seem like a waste, however, you still have a fairly high chance of getting some other legend that you didn’t already roll.
the general idea is 1 legend every 2 packs.
legend can be placed anywhere, on average will be at spot 5.5 out of the ten available.
thus 4.5 “wasted” rolls per 20 rolls, when buying packs.
a regular roll has 5% chance of legend, a “wasted roll” has 4.25% chance of legend.
also on average 1 legend every 2.3 rare eggs.
lets take 80 rolls as an example.
single rolling nets you 4 legendaries.
pack rolling nets you 4 rare gems → 4/5/2.3 = .4 of a legendary plus
about 20 “wasted” rolls (4.54) thus .042520 = .8 of a legendary.
Adding the 6 not wasted packs → 3 legendaries.
Thus a pack gives you 3 + .8 + .4 legendary = 4.2 legendary versus 4 legendaries.
This was a general example, but I also wrote some code that calculates whether its better to roll a single or a pack, based upon how many legends/myths you have from the banner, the non-banner legends/myths, and rare egg legends/myths. You can find it here 3wq8cuz82 - Python - OneCompiler, and it’s easily customizable if u you want to try it for yourself.
If you rightfully don’t trust random links on the internet, I also made a few general calculations.
I love number crunching, possibly as much as you. Looks like great work you’ve done but I’m confused about some of the numbers you’re using…
Your assumption of exactly 1 legendary every 20 rolls is giving you some weird results here, along with your classification of “wasted rolls”. I mean, what exactly are you doing with single rolls here. How many gems will you spend for the 4 legendaries? You’ve gone down this path of assuming for packs you will always get the legendary in the second of each pair of packs. You also got the incorrect number of rare gems. From 80 rolls you should have 8 rare gems.
If you correct for the number of rare gems then the final comparison should read “80 gems in packs = 4.6 legendaries” vs “??? gems in singles = 4 legendaries”. The ??? could be something like 40-60 gems. You can’t tell and it makes this conclusion a bit dubious.
I’m sorry to criticise but you scaled up your 20 rolls example in a way that doesn’t work and made a mistake along the way.
I think it could be looked at in a much simpler way…
Chance for a featured legendary/mythic in a special egg = 3%.
To get one it takes, on average, 33.3 eggs (133.3 gems).
Doing it with packs means you will, on average, “waste” 4.5 eggs (18 gems) but get 3.33 rare gems.
There’s a ~10% chance of these extra eggs hatching you one of the remaining featured so call that “4.1 wasted eggs” on average (lowered by ~10%). (= 16.36 gems)
So to get a featured legendary/mythic with singles it costs 133.3 gems and with packs it costs 149.7 gems (getting an additional 3.33 rare gems as a bonus).
The real question is… does 16.36 gems = 3.33 rare gems in value??
I’ve looked at this in detail before and I’d say 1 rare gem should equal 2 - 2.5 rare gems (the lower number as you get a larger collection).
Hence 16.36 / 2 = 8.18 rare gems worth of value. 3.33 rare gems isn’t anywhere close.
Now there are lots of ways of looking at this and I’ve taken the case of a person not having any featured and only caring about hatching one before stopping which means it’s more on the extreme end of not valuing these “wasted gems”. However, I hope it’s clear enough to see where people are getting the conclusion that the gems you save from singles are worth more than the rare gems you’d otherwise have. The rare gems only become better if you hatch something like 8+ packs in an egg.
what I was trying to highlight was that in the long run, if all you care about is having the highest quantity of legendaries/mythic, then hatching packs is more efficient, even if you have “wasted” rolls as the rare gems make up for it until you essentially have every rare gem legendary mythic, and was trying to see if there was a threshold if at a certain point the rare eggs lost value.
I agree with your analysis of the singles always getting to a featured legendary with fewer rolls.
Also, forgive my sloppiness in the calculation in my first post as you rightly pointed out a few mistakes I made.
I realize what I had overlooked is the low demand on legends from the rare egg hatch as the probability you get a monster that is useful to your team is very a lot lower compared to the extra hatches saved by investing in singles in banners you are interested in.
Binomial probability is applied when the outcome is either this or that strictly .That’s why its called "bi"nomial…… Egg hatches are quiet different since there are 4 outcomes now. If u apply binomial for neo-egg system , the numbers or the % u calculate for each rarity will overlap with success rate of other 3 rarity . That overlap will be the error in calculation that you and some others have constantly posted in forum/line from some random “Binomial calculator” link… Now that has annoyed me forever!
You don’t even have to take the extrem end tough even with 3/4 and 2/4 desired feat. pulls it’s still better. With 2/4 it’s getting close tough.
So unless you are whaling like a madman for a single feat. AND are willing to go real deep for it, it’s always slightly worse. I am not taking time into any consideration here because then it gets crazy.
all I’m saying is you don’t find your keys and then keep looking 
