[Theory] Coloured source count for casting Cryptic Command

The math regarding Cryptic Command and U sources eludes me. Here, it states in order to have a 90% cast success rate in relation to U count, one needs 22 sources of U for a T4 requirement of UUU.

This multivariate hypergeometric distribution calculator, the math of such explained here, deems that 4 Cryptic Commands require 24 sources of U for a T4 Cryptic Command. Which is quite surprising, as typically, for high card counts the calculated source count for a 90% success rate is quite similar between the Frank Analysis tables and the multivariate hypergeometic distribution tables, but there's a difference of two here.

Even more confusing, I've seen competitive lists from professional esteemed players, playing anywhere between 16-22 U counts when running 4 Cryptics. Granted, the 16 U list was running 4 Aether Vials alongside Eternal Witness(softlock with cryptics), but it's apparently common for even Jeskai control to run 19 U for 4 Cryptics.

I don't understand why this is the case, and I'd like to generate a discussion on it.

Does card-draw from cards like Remand, slightly reduce at all the need for U counts? I see that drawing cards increases our likely hood of drawing a U source, but it also increases our likelihood of drawing a Cryptic, so I'm unsure about this.

I understand that, we won't have 4 lands by T4 90% of the time, anyway. Do we need less coloured sources to support the high-end part of our curve? I have been thinking the opposite, that in order to have our higher end of our curve not get clogged in our hand and be worthwhile to play, we must be able to consisently cast it on time.

Can it simply be that the lists that play 4 Cryptic Commands, gain enough value by such, and are slow enough, that they're okay with not playing it consistently on T4? That the trade off for playing more U sources, or less Cryptics, simply isn't worth it, for whatever reason.

What skews the hypergeometric distribution, and why most cryptic decks can get away with only 20-22 blue sources, is the added draws that happen between turns 1-4. With the calculation you are only including 11 instances (7 card opener w/o mulligan and 4 draw steps.) Literally *zero* cryptic decks are built without additional draws planned in turns 1-4. Serum Visions is almost guaranteed to be a 4-of (and skews the math even further because of the scry factor, you actually have 'control' of 2 more draws) and most will play another source of draw as well (Remand, Electrolyze, Think Twice, Esper Charm, etc.) This is where your math gets complicated, but regardless I tend to go with historical precedent (I copied mana-bases from Shaun McLaren and Gregory Orange.)

For instance, in my old UWR control deck I played 26 lands in this mix:

4x Celestial Colonnade
3x Arid Mesa
3x Flooded Strand
2x Scalding Tarn
2x Hallowed Fountain
2x Steam Vents
1x Sacred Foundry
1x Sulfur Falls
3x Island
1x Plains
1x Mountain
3x Tectonic Edge

That's 20 blue sources of mana (gets complicated again when using fetchlands, because when you fetch you are actually taking 2 sources of blue out of your library each time.) I played a full set of Electrolyze and between 2-4 Remand. I was almost always drawing an extra card before turn 4. Another part of the equation is whether or not you *absolutely* need to cast Cryptic Command on turn 4. The answer most often is no. It was really a turn 5-7 card, with many of the other powerful cards like Snapcaster Mage + 1-3mana spell doing the same job as Cryptic.

TL;DR - Cryptic isn't a crucial card to play by turn 4 (not like Splinter Twin was) and you actually have redundant Cryptic-like effects (Snapcaster Mage) which skews how often you'll be able to accomplish/need UUUX exactly on turn 4. Hypergeometric distribution is an excellent tool, and a great place to start, but it won't solve all your deck-building challenges. After you get a rough start with HD you need to then start making decisions based on critical turns and critical cards to include in your deck. And then there's always the easy route: look at what the pros are doing, and do that.

I hope I didn't come off as pedantic, that wasn't my intention. This is something I am learning as well.

Good luck!

Serum Visions is a really good way to skew the distributions. If your only goal with SV is to find 1 blue source, it effectively digs 3 cards down for you. Remand and other draw effects help too, but scrying is very powerful.

Some decks run cards like Sunken Ruins to help out a lot. Lands like that basically turn a basic swamp into an island.

That and like is being said, Cryptic is still good even if it isn't cast on turn 4.