Introduction
Thanks to the Balance Math Managers on the Discord and everyone who helped with the math and contributing almost 500 datasets  you are awesome!
This spreadsheet is where we gathered all the datasets.
And remember, this is not data from the actual game, this is made using several test cases that were made in regular matches, but we have no real way of checking what the code of the game is actually doing.
Overview
Remember the good old Balance when you could arrange your Deck in a particular way to guarantee you a specific number of Monsters, Spells and Traps in your starting hand? There's no way to do that anymore, with the new Balance mechanic, the best you can do is guarantee a minimum and a maximum.
With the new Balance we can’t be so sure in exactly how many cards of a type we will open with, we now have a table of probabilities of how many of a type can you open given how many you have in the Deck.
It’s not that difficult, here we have the table of Number of Cards in the Deck x Number of cards you open (this table only works for a 20 card Deck):
As you can see from the table, no quantity of cards in the Deck gives you a 100% chance of opening with an exact number of cards like 5, 10 or 15 did before, giving you 1, 2 and 3 cards in your starting hand respectively.
But we can get some interesting results out of that like with 6 cards of a type in the Deck, you are guaranteed AT LEAST 1 of it in your starting hand. With that in mind, we can get what is the maximum and minimum cards of a type you can open with and gather it all up, which looks something like this:
So, for example, if you have 12 Monsters in your Deck, you can only open with 2 or 3 Monsters, never with 1 and never with 4!
Examples

A Deck with 5 Monsters, 7 Spells and 8 Traps:
 You are guaranteed 1 Spell and 1 Trap since in the Deck you have 6 or more of them.
 You cannot get more than 2 Spells or 2 Traps.
 You are not guaranteed a Monster in your hand, but you can open with it about 60% of the time (check Table 1). You can very rarely open with 2 Monsters.

A Deck with 13 Monsters, 3 Spells and 4 Traps:
 Even though you don’t have 6 Traps nor 6 Spells, you are guaranteed 1 Trap or Spell since with 13 Monsters you can’t get 4 Monsters in your starting hand, so the slots missing are filled with Spell/Traps.
Top Deck
You feel like your top Deck is getting influenced by Balance? You open without any Spells but you feel like every time it happens, there is a Spell in the top of your Deck… You're right! Balance actually influences your top Deck.
Why we think that? Look at these example stats carefully: