How to Use This Tool
Follow these steps to calculate card draw probabilities for your game:
- Select your deck type from the dropdown to auto-fill total deck size, or choose "Custom" to enter your own value.
- Enter the number of target cards (the specific cards you want to draw) present in your deck.
- Enter how many cards you will draw from the deck.
- Select the probability type: exact number of targets, at least X targets, or at most X targets.
- Enter the target draw count (X) matching your probability type selection.
- Click the "Calculate Probability" button to see your results.
- Use the "Reset" button to clear all inputs and start over, or "Copy Results" to save your breakdown.
Formula and Logic
This tool uses the hypergeometric distribution to calculate probabilities for draws without replacement, which applies to most card games and gacha systems where items are not returned to the pool after being drawn.
The core formula for the probability of drawing exactly k target cards is:
P(X = k) = [C(K, k) * C(N-K, n-k)] / C(N, n)
Where:
- N = Total number of cards in the deck (population size)
- K = Number of target cards in the deck (success states in population)
- n = Number of cards drawn (sample size)
- k = Number of target cards drawn (success states in sample)
- C(a, b) = Number of combinations of a items taken b at a time
For "at least X" calculations, the tool sums probabilities for all k values from X to the maximum possible targets you can draw. For "at most X" calculations, it sums probabilities from 0 to X.
Practical Notes
Keep these gaming-specific factors in mind when using this tool:
- Many TCGs like Magic: The Gathering or Yu-Gi-Oh! have deck size minimums (60 for MTG, 40 for Yu-Gi-Oh!) that you should account for in your total deck size.
- Gacha games often use "pity" systems that alter base probabilities after a certain number of pulls, which this tool does not account for (use the "Custom" deck type to adjust for modified probabilities if needed).
- Some games use draws with replacement (like certain video game card pulls that reshuffle after each draw) — this tool calculates without replacement, so adjust your total deck size to 1 for independent gacha pulls with replacement.
- Competitive players should note that meta shifts (like banning certain cards) will change the number of target cards in your effective deck, so update your inputs accordingly.
- Streamers calculating odds for on-stream pulls should remember that RNG is independent per draw, so past results do not affect future probabilities.
Why This Tool Is Useful
Gamers, designers, and streamers save time and reduce errors by using this tool instead of manual hypergeometric calculations:
- TCG players can plan mulligans, sideboard choices, and win conditions based on accurate draw odds.
- Game designers can balance card rarities and deck compositions by testing probability scenarios during development.
- Streamers and content creators can quickly generate accurate probability breakdowns for viewer questions or challenge runs.
- Competitive players can verify if a strategy relying on drawing specific cards is mathematically viable before investing time in testing.
Frequently Asked Questions
Does this tool work for gacha game pulls?
Yes, but you will need to adjust inputs for gacha-specific mechanics. For single-pull gacha with replacement, set total deck size to 1 and target cards to the number of "winning" items in the pool. For 10-pulls without replacement, set total deck size to the gacha pool size and cards drawn to 10.
Why is my probability showing as 0%?
This usually means your inputs are impossible: for example, if you enter 5 target cards in a 60-card deck, draw 3 cards, and ask for at least 6 target cards. Check that your target draw count does not exceed the number of cards drawn or the number of target cards in the deck.
Can I calculate odds for multiple target card types?
This tool calculates probabilities for a single target card type (e.g., "all 3-cost cards" or "all rare cards"). To calculate odds for multiple distinct types, run separate calculations for each type and combine the results manually if needed.
Additional Guidance
For best results, always verify your inputs against your game's current rules and meta. Patch updates to TCGs or video games can change deck size limits, card pools, or gacha rates, which will affect your probability calculations. If you are calculating odds for a custom game, test your results with small sample sizes first to confirm the tool's output matches your expectations. Save your result breakdowns using the copy button to reference later when adjusting your strategy or game design.