TCG opening-hand probability: at least one copy
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To calculate the chance of at least one target card in an opening hand, count the deck, the target copies and the cards drawn without replacement. The answer is one minus the chance of drawing no target copies. The calculator below implements that model; it does not simulate mulligans or predict wins.
This is useful when comparing several legal deck configurations or checking an intuition such as four copies should be almost certain. A transparent calculation helps you see the actual difference without claiming that the higher percentage automatically creates the better deck.
Define what counts as a successful hand
Start with a specific event: at least one copy of this card among the first seven cards, for example. That event is different from finding two copies, a combination of two named cards, or a card plus the resources required to use it.
Let N be the number of cards in the deck being sampled, K the number that satisfy your target condition, and h the number drawn. The model assumes each possible group of h cards is equally likely after an adequate shuffle and that drawn cards are not returned during that sample.
Count only the deck from which the hand is drawn. Leaders, tokens, separate resource decks and other components do not belong in N merely because they are part of your game setup. Use your game's current rules to establish the actual opening-hand size and setup procedure.
Use the chance of missing every copy
The first draw misses with probability (N − K) / N. If it misses, the next draw misses with probability (N − K − 1) / (N − 1). Continue through h draws and multiply those conditional probabilities. Subtract the result from one.
Written with combinations, the same result is P(at least one) = 1 − C(N − K, h) / C(N, h), where C(a, b) counts the possible groups of b cards from a. If h is larger than the number of non-target cards, missing every target is impossible, so the answer is 100%.
This is an application of sampling a finite population without replacement, described by the hypergeometric distribution. The calculator uses the sequential product so you do not need to enter huge factorials into a spreadsheet. Its percentages are calculated from your inputs, not from measured gameplay data.
A small worked example you can check by hand
Imagine a deliberately tiny hypothetical deck of five cards, with two target cards. Draw two. There are ten equally likely two-card groups: five choose two. Three groups contain only non-target cards because there are three non-target cards and three choose two is three.
The chance of missing is therefore 3/10, and the chance of at least one target is 7/10, or 70%. The sequential version gives the same result: miss the first draw with 3/5, then miss the second with 2/4. Their product is 30%, leaving 70%.
This example also shows why simply multiplying the first-draw probability by the hand size does not work. The successful groups overlap if you count each target occurrence independently. A hand with both targets is still one successful hand for the event at least one.
| Hypothetical deck N | Target copies K | Draws h | Chance of at least one |
|---|---|---|---|
| 5 | 2 | 2 | 70% |
| 10 | 1 | 3 | 30% |
| 10 | 0 | 3 | 0% |
| 10 | 10 | 3 | 100% |
| 10 | 2 | 0 | 0% |
Several cards can form one target category
If either of two different cards independently satisfies the same condition, you can count their combined copies in K. For example, three copies of one card and two of another create five targets, provided every one independently meets the event you are measuring and none has been counted twice.
That does not calculate a two-card combo. A hand containing card A but no card B succeeds for at least one of A or B, yet fails for having both A and B. Calling those two events the same thing can produce a reassuring percentage for a combo the model never checked.
Likewise, a search effect is not necessarily equivalent to another direct copy. It may need resources, a legal target, a timing window or another card to function. You can model a simplified target category, but label the simplification instead of presenting its result as the exact chance of a playable opening.
Mulligans and later draws need a different question
An opening hand after a mulligan depends on the game's actual procedure and your decision rule. A full reshuffle, a partial replacement and a bottom-of-deck selection create different models. The calculator reports the initial sample described by N, K and h; it does not know which hands you would keep.
Later draws also depend on what is already known. If no target has appeared among seven drawn cards, the remaining deck has fewer cards but the same target count, assuming no other zone movement. A new conditional calculation should use that remaining state rather than silently adding independent probabilities.
Known cards removed before a draw can change the available targets. Set-up procedures, revealed information and separate zones must be incorporated according to the game and the question. Do not use the same percentage for every TCG merely because two decks share a printed size.
Interpret the difference before changing a deck
Increasing K usually increases the chance of seeing a target, but it also replaces something else in a fixed-size deck. The calculation says nothing about the displaced card's function, redundant copies later in the game or the resources needed to play the target.
Record each tested configuration and the event you measured. Compare percentage-point changes, then consider whether those changes serve your plan. A legal deck with a higher opening-hand percentage may still be worse at another task you care about.
The result is a probability under stated assumptions, not a promise about the next match. Missing a likely card in one game is compatible with the model. To investigate suspicious physical draws, examine shuffling and records rather than treating one memorable hand as proof that the arithmetic failed.
Opening-hand calculator
Frequently asked questions
Why is the deck size reduced after a missed draw?
Because sampling is without replacement. The drawn non-target card is no longer in the deck, so both the remaining total and the remaining non-target count fall by one.
Can this calculate a booster-pack pull rate?
No. A known deck sampled without replacement is different from a product with an unknown collation process. Do not enter guessed rarity quantities and present the result as an official pack probability.
Does at least one include hands with two copies?
Yes. One, two or more targets all satisfy this event. Exactly one is a different calculation, as is requiring multiple distinct cards.