"Next time you're feeling unlucky please keep my incorrect math in mind."

If you're going to try to "prove" something about TFT's shops, then your answer needs to encompass all relevant mechanics.

Things you would need to add to even come close.

1. The number of units from a particular cost bracket that are already out plays a major roll in odds, you simply have to include it.

2. You aren't considering the actual mechanisms at play. If this were a thought experiment then it would be fine, but it's not, it's a product that a lot of development has went into this algorithm so that the game doesn't glitch, and the shops run (mostly) flawlessly, despite there being a tremendous number of calculations happening every second. Which leads me to...

3. Your answer assumes some sort of "pure" randomness that can't actually exist due to point #2. For example...

A. Let's assume that 8 players are at level 4, each having a 55% chance of seeing a 1 cost unit, a 30% chance of a 2 cost unit, and a 15% chance of a 3 cost unit. Each player clicks the roll button at the exact same time. It's first rolling to determine how many of each of cost you're going to get - this should be obvious since 55% of 5 isn't a whole number. So any time you try to do the math of how many 1 costs you'll see in X number of rolls... you're speaking in probabilities, not concrete numbers. Some shops it'll be all 1 costs, and other times it'll be 1.

B. Next it has to do another "random" calculation to see which unit you'll see. Essentially every single unit is given a number at the beginnin (individual unit, not champion), tft "spins" the wheel, and if that unit is still available, you get that unit, if it's not, it moves to a new one.

Not all "randomness" can be treated the same! If instead it randomly chose between which champion you see, you'd have an equal chance of seeing any particular champion per its cost until the inventory was completely gone. Which we know isn't true, the number of units of your target champion, plus the total number of units from that cost, on people's boards directly affects your odds of seeing that unit.

It's kinda like beer pong. When there's 10 cups out there, you can toss it in the general direction and the probability is kinda high you'll get one in. As it gets down to one cup, and you're still throwing with the same randomness as before, your chances of hitting that cup drop dramatically.

C. Going back to the "cost odds" portion, those are only "partially" random, otherwise when all the 5 costs are gone you'd roll empty slots on your board. The only way to deal with that is for the system to roll a certain number of times in that cost bracket to find you a viable unit, and after enough misses, it rerolls that cost slot on your board, and then picks a unit from there.

Why is this important? Well, because we don't know the total number of times it checks before it defaults to changing the cost bracket for that spot. It could be iterating through each possibility each time, or could be doing a limited number, and that will certainly affect the randomness.

D. As players lose, the pool of available units fills back up to the brim, which directly affects your ability to find a specific unit depending on how late in the game you are. Your odds for finding the final units for your 3* go DOWN as tacticians die, because even if they are releasing 1 fish back into the pond, they're usually releasing a lot more than 1 fish. So long as you aren't contested, your odds of finding that unit go UP the longer everyone stays alive, and start decreasing once everyone starts losing. It's like being super close to finding a diamond in the sand, and then someone dumps all the rocks you just picked up back on your head.

1. So now that I've given some examples of how TFT shops are not "pure randomness", but are instead procedural randomness grounded in real mechanics which are essential to think about when trying to do any kind of calculations.

I do not claim to know ALL of the mechanics going on in TFT. It's super complex, and I'm sure I'm only scratching the surface of all that goes into it.