Yu-Gi-Oh! is a game of opening hands — most decks live or die on whether their five cards can start the combo. This calculator uses hypergeometric math for single cards and the multivariate hypergeometric distribution for two-card combos, so you can see exactly how often your deck delivers what it needs on turn one.
How it works
For a single card, with deck size N, copies K, and hand size h, the chance of opening at least
one copy is:
P(at least 1) = 1 - C(N - K, h) / C(N, h)For a two-card combo with disjoint pieces (counts K1 and K2), the tool uses inclusion and
exclusion over the multivariate hypergeometric distribution:
P(both) = 1 - P(miss A) - P(miss B) + P(miss both)
= 1 - C(N-K1, h)/C(N, h) - C(N-K2, h)/C(N, h) + C(N-K1-K2, h)/C(N, h)This correctly handles the fact that drawing one piece changes the odds for the other, because the two groups share the same finite deck. The pieces must be distinct cards, so their combined copies cannot exceed the deck size.
Example and notes
In a 40-card deck running 3 copies of a starter, the chance of opening at least one in a 5-card hand is about 33.8%. If your combo needs one of a 3-of and one of a different 3-of, the chance of opening both in five cards drops to roughly 9.5% — a stark illustration of why consistent decks lean on searchers and “any one of many” starters rather than rigid two-card combos.
Keep in mind:
- Going second adds a sixth card; bump the hand size to 6 to see the improvement.
- The combo mode assumes the two pieces are different cards. For “any one of several interchangeable starters”, add their copies together and use single-card mode.
- Search cards, draw spells, and hand traps are not modelled — they only raise your real consistency above these baseline figures.