How to Use This Tool
Follow these steps to calculate card draw probabilities for your game:
- Enter the total size of your deck (e.g 60 for Magic: The Gathering, 52 for standard poker).
- Input how many copies of your target card are in the deck (e.g 4 for a playset of a TCG card).
- Specify how many cards you are drawing (e.g 7 for an opening hand, 1 for a single draw).
- Set the number of target card successes you want to calculate for.
- Select whether you want the probability of exactly that many successes, at least that many, or at most that many.
- Click the Calculate button to see your full probability breakdown.
- Use the Reset button to clear all inputs and start over, or Copy Results to save your breakdown.
Formula and Logic
This calculator uses the hypergeometric distribution, which models the probability of k successes in n draws without replacement from a finite population of size N containing exactly K successes.
The core formula for exactly k successes is:
P(X = k) = [C(K, k) * C(N - K, n - k)] / C(N, n)
Where C(a, b) is the combination function, calculating the number of ways to choose b items from a items without regard to order.
For "at least k" probabilities, we sum the exact probability for all values from k to the maximum possible successes (min(K, n)). For "at most k" probabilities, we sum from 0 to k.
Practical Notes
These tips apply to real-world gaming scenarios for this tool:
- TCG and tabletop games often have deck size limits (e.g 60 minimum for Magic: The Gathering, 50 for Pokémon TCG) that you should follow when inputting values.
- Video game RNG card draws (e.g Genshin Impact's gacha, Slay the Spire card rewards) may use modified hypergeometric distributions or hidden weighting, so results are estimates for unweighted systems.
- Meta variations: If a game has banned cards or adjusted deck limits, update your target card count and deck size accordingly.
- Patch-dependent values: For live-service games, check if recent patches changed deck size limits or draw mechanics before using old values.
- Mulligan rules: If your game allows redrawing hands, calculate probabilities for the mulliganed hand size separately and combine results for full accuracy.
Why This Tool Is Useful
Gamers, designers, and streamers benefit from this tool in multiple ways:
- TCG players can optimize deck builds by calculating the odds of drawing key cards in their opening hand.
- Game designers can balance card rarities and draw mechanics to ensure fair, predictable RNG for players.
- Streamers can use real-time probability breakdowns to explain odds to viewers during live gameplay.
- Competitive players can plan mulligan strategies based on the likelihood of improving their hand.
- Board game designers can test if card draw probabilities align with intended game pacing and difficulty.
Frequently Asked Questions
What deck size should I use for a standard TCG?
Standard TCG deck sizes vary: Magic: The Gathering uses 60-card minimum decks, Pokémon TCG uses 60-card decks, Yu-Gi-Oh! uses 40-60 card decks. Always use your actual deck size including all cards, not just the minimum limit.
Does this work for video game gacha or card draws?
This tool works for video game card draws that use true random without replacement (like Slay the Spire's card rewards). Many gacha games use weighted RNG or pity systems that this calculator does not account for, so results are only accurate for unweighted, replacement-free draws.
How do I calculate probabilities for multiple target cards?
If you want the probability of drawing any of multiple different target cards, sum the number of copies of all target cards to get your total target card count. For example, if you want the odds of drawing any of 2 copies of Card A and 2 copies of Card B, enter 4 as your target card count.
Additional Guidance
For more accurate results in competitive play, always use your exact current deck size, including any cards you have already drawn or discarded if calculating mid-game probabilities.
When balancing game systems, aim for key card draw probabilities between 60-80% for opening hands to keep gameplay consistent without feeling too RNG-dependent.
If you are calculating probabilities for a game with replacement (e.g drawing a card, then shuffling it back), use a binomial distribution calculator instead, as this tool is for draws without replacement only.