Yugioh probability calculator
In the yugioh probability calculator, 3 copies of a card in a 40-card deck give a 33.76% chance of at least one in a 5-card opening hand.
P(X ≥ 1)
- Probability of at least 1 success 5 drawn from 40 without replacement, 3 of which are successes.
- 33.76%
- Formula
- 1 - P(X = 0) = 1 - C(37, 5) / C(40, 5) = 1 - 435,897 / 658,008
- As a fraction Counted exactly in whole numbers and reduced: the draws that qualify over every possible draw.
- 667 / 1,976
- Probability of fewer than 1 The complement: 1 minus the answer above.
- 66.24%
- Mean (expected successes) n x K / N = 5 x 3 / 40.
- 0.375
- Variance n x (K/N) x (1 - K/N) x (N - n)/(N - 1). The last factor is the finite population correction: the variance of drawing with replacement, shrunk because each draw changes what is left.
- 0.3113
- Standard deviation
- 0.5579
- Most likely count
- 0 (66.24%)
- Possible counts At most the smaller of the draws and the successes, and at least whatever the failures cannot fill.
- 0 to 3
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|---|---|---|
| 0 | 0.6624 | 0.6624 | 1 |
| 1 | 0.3011 | 0.9636 | 0.3376 |
| 2 | 0.03543 | 0.999 | 0.03644 |
| 3 | 0.001012 | 1 | 0.001012 |
- Probabilities in the table have four significant figures. Terms smaller than 10^-300 of the most likely one are treated as 0.
How the Hypergeometric Calculator works
Drawing from a deck is a draw without replacement, so the odds of seeing a card are hypergeometric. With N cards in the deck, K copies of the card you want and n cards drawn, the chance of at least one copy is P = 1 - C(N - K, n) / C(N, n), where C(a, b) counts the ways to choose b cards from a. The fraction is the share of hands made entirely of other cards.
Worked example: a 40-card Yu-Gi-Oh! deck with 3 copies and a 5-card opening hand. C(37, 5) = 435,897 hands miss every copy, out of C(40, 5) = 658,008, so P = 1 - 435,897 / 658,008 = 33.76%, exactly 667/1,976. Going second adds a sixth card and raises it to 39.43%.
Copies matter most. In the same 5-card hand, 1 copy is 12.5%, 2 copies 23.72% and 3 copies 33.76%. Counting several different cards that do the same job as one target works the same way: 6 such cards give 57.71%, and 9 give 74.18%.
Deck size works against you. The same 3 copies in a 60-card deck open in only 23.33% of 5-card hands, since every extra card is one more way to draw something else.
Opening two copies is rare: at least 2 of 3 in a 5-card hand from 40 is 3.64%, or 9/247.
The method is the same in any card game. Magic: The Gathering uses 60-card decks, up to 4 copies and a 7-card opening hand, which gives 39.95% for at least one copy.
Hypergeometric Calculator FAQ
What are the odds of opening a 3-of in a 40-card deck?
33.76% in a 5-card hand going first, and 39.43% with the sixth card going second.
How does a card probability calculator work?
It counts hands. The chance of at least one copy is 1 minus the number of hands with no copies, C(N - K, n), divided by all possible hands, C(N, n).
Does a bigger deck lower the odds of drawing a card?
Yes. 3 copies in a 5-card hand are 33.76% from 40 cards and 23.33% from 60.
How do I find the odds of drawing one of several cards?
Add their copies together and use that total as K. Three cards at 3 copies each make K = 9, which opens at least one of them 74.18% of the time from 40 cards.