The math behind your deck (and why a gut feeling isn't good enough)

Your deck is a fixed set of 60 cards. Every hand you draw is a sample taken from that set, without putting anything back. That single fact — no replacement — is what makes the math behind deckbuilding a specific, well-defined thing called hypergeometric probability, and it's the same math that answers "what are my odds of drawing a specific card" whether you're checking your opening hand or a topdeck on turn 8.

You don't need to do this by hand. TCG Tools runs it automatically against your actual decklist. But understanding what it's actually calculating makes you a sharper deckbuilder — and it turns "I feel like this deck is consistent" into something you can check in seconds.

The question, precisely

"What's the probability I have at least one of my 4 copies of a card in my opening 7-card hand?"

That's not a vague question — it has one exact answer, because you know everything that matters: 60 cards total, 4 copies of the card, 7 cards drawn, no replacement. Plug those into the hypergeometric distribution and there's nothing left to guess.

The answer: about 40%.

If that feels low, that's the point. A card you're relying on to open your hand is, more often than not, not in your opening hand. That's not bad luck — that's just what 4 copies in 60 cards actually gets you. If your gameplan needs that card early, either you need more copies, more ways to find it, or a plan that survives not having it turn 1.

The mulligan version of the same question

Run the same math on your basic Pokémon count instead of a single card, and you get your actual mulligan rate — not a guess, the real number.

With 9 basic Pokémon in a 60-card deck, the odds of a hand with zero basics — a mulligan — sit at roughly 30%.

That's nearly one hand in three. Drop to 7 basics and that number climbs noticeably; add a couple more and it falls. This is exactly the kind of thing that's cheap to fix at deckbuilding time and expensive to discover at a tournament.

Why "I playtested it and it felt fine" doesn't settle this

Ten test hands is not a sample size that can tell a 40% card from a 55% card — the noise is bigger than the signal. Human memory doesn't help either: a run of three bad hands in a row feels like a pattern even when it's exactly what you'd expect from ordinary variance. The math doesn't get fooled by a bad session or a lucky one, because it's not measuring a session — it's measuring the deck itself.

This is also why "combo probability" — the odds of assembling a specific multi-card setup by a given turn — matters more than any single card's odds in isolation. A two-piece combo where each piece is individually likely can still be surprisingly unlikely together, for the same reason two 80% chances multiplied out are only 64%, not 80%.

The actual formula, if you want it

For the curious: the probability of drawing exactly k copies of a card, from K copies in a deck of N cards, in a hand of n cards, is:

P(X = k) = [C(K, k) × C(N−K, n−k)] / C(N, n)

where C(a, b) is "a choose b." For "at least one," it's usually easier to calculate the zero case and subtract from 1 — which is exactly how the 40% and 30% numbers above were worked out.

You can absolutely do this by hand with a calculator and a combinatorics reference. Or you can paste a decklist into TCG Tools and get every one of these numbers — consistency score, mulligan odds, combo probability — without doing the arithmetic yourself. Either way, now you know what the number actually means when you see it.