Card draw odds
How likely are you to draw the cards you need? Set the deck, the cards you're looking for and how many you draw, and get the exact chance in your opening hand and on every turn after it.
| When | Cards seen | Chance | Odds |
|---|---|---|---|
| Opening hand | 7 | 39.9% | about 2 in 5 |
| Turn 1 | 8 | 44.5% | about 3 in 7 |
| Turn 2 | 9 | 48.8% | about 1 in 2 |
| Turn 3 | 10 | 52.8% | about 6 in 11 |
| Turn 4 | 11 | 56.6% | about 4 in 7 |
| Turn 5 | 12 | 60.1% | about 3 in 5 |
| Turn 6 | 13 | 63.4% | about 5 in 8 |
How to use it
Enter the size of the deck and how many cards you draw: the opening hand first, then the cards drawn each turn. Under Cards you're looking for, add a group for each thing you need, such as a 60-card deck with 4 copies of a key card. A group can also be several different cards that do the same job: 4 of one and 2 of another make a group of 6.
With more than one group, the big plate is the chance of meeting every requirement in the same hand, and the list below it shows each group on its own. The table carries on turn by turn; if your game skips the first draw, read one row up.
The maths
Drawing without putting cards back follows the hypergeometric distribution: count the ways to pick the cards you want, times the ways to pick the rest of the hand from everything else, and divide by the number of possible hands. Several groups multiply one such count per group. The deck is assumed to be well shuffled.
Sources
- Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd edition (American Mathematical Society), §5.1 “Important Distributions”: the hypergeometric distribution. Free GNU FDL edition on Dartmouth's site.
- NIST Digital Library of Mathematical Functions, §26.3 “Lattice Paths: Binomial Coefficients”: counting the ways to choose cards.
Questions
Why isn't the chance just 4 × 7 ÷ 60?
That sum, about 0.47, is the average number of copies you'd hold, not the chance of holding one. Some hands have two or three copies and many have none, so the chance of at least one copy in seven cards from sixty is 39.9%.
What does “at most” help with?
Checking for too much of something: say, no more than 4 of your 24 resource cards in the opening hand. Combine it with an at-least group to ask for a hand that is neither short nor flooded.
Does it handle mulligans?
Not directly, because the rules differ from game to game. A full reshuffle gives a fresh hand with the same odds again, so the chance of failing twice in a row is the chance of failing once, squared.