In competitive Pokémon Trading Card Game (TCG) play, games are rarely decided solely by top-deck luck. Rather, high-level players rely on mathematical probability to build decks that consistently execute their setup strategy on Turn 1.
1. The Fundamental Formula: Hypergeometric Distribution
When drawing a 7-card opening hand from a 60-card deck without replacement, the exact probability of drawing $k$ copies of a specific card group is given by the hypergeometric probability mass function:
Where:
- N = 60: Total deck size at the start of the game.
- n = 7: Opening hand size.
- K: Total number of target cards in your 60-card deck (e.g., number of Basic Pokémon or Supporters).
- k: Number of target cards you want in your opening hand.
2. Mulligan Probability: How Many Basic Pokémon Do You Need?
A mulligan occurs when your 7 opening cards contain zero Basic Pokémon. The probability of having to mulligan is simply:
Here is the exact mulligan probability relative to your Basic Pokémon count ($B$):
| Basic Pokémon Count | Mulligan Probability | Setup Reliability |
|---|---|---|
| 4 Basics | 59.0% | Extremely Risky |
| 6 Basics | 44.2% | High Mulligan Risk |
| 8 Basics | 32.6% | Competitive Standard |
| 10 Basics | 23.6% | Very Reliable |
| 12 Basics | 16.7% | Ultra-Consistent Engine |
3. Monte Carlo Simulation vs. Static Math
While hypergeometric formulas calculate individual card draws cleanly, real Pokémon TCG scenarios involve complex dependencies—such as mulligan reshuffling, prize card placement, and conditional search chains (e.g. Arven into Buddy-Buddy Poffin into two Basics). That is why our Monte Carlo Simulator runs 100,000 complete games to account for these real-game interactions.