TCG Opening Hand Probability Calculator (MTG, Pokémon, Yu-Gi-Oh, Commander)

How it works

hypergeometric: P(exactly k) = C(K,k) × C(N−K, n−k) ÷ C(N, n) · P(at least k) = Σ from k · cards seen by turn T = hand + T − 1 (on the play) or hand + T (on the draw) · London mulligan m times: 1 − (1 − p)^(m+1) · copies needed = smallest K with P ≥ target

Drawing cards from a shuffled deck without replacement is the textbook hypergeometric situation, so the chance of seeing a given card depends only on how many copies the deck holds, how big it is and how many cards have been looked at — the opening hand plus a draw per turn, one fewer for the player who goes first. The numbers explain the shape of competitive decks: four copies of anything is still a coin flip by the mid-game, which is why builders count "functional copies", pad key effects with redundancy and draw, and why a singleton format like Commander leans on tutors rather than any one card.

Worked example

A 4-of in a 60-card Magic deck shows up in the opening seven 39.9% of the time, 48.8% by turn 3 on the play (9 cards seen) and 52.8% by turn 4; one London mulligan lifts the opener to 63.9%. To see the effect in 90% of turn-3 hands you need about 13 copies' worth of redundancy, and even 8 copies only reach 75%.

Frequently asked questions