Opening hand probability calculator
Deck characteristics
Results
Chance of drawing it
39.9%
| Copies in your hand | Exactly | At least |
|---|---|---|
| 0 | 60.1% | 100% |
| 1 | 33.6% | 39.9% |
| 2 | 5.9% | 6.3% |
| 3 | 0.4% | 0.4% |
| 4 | 0.0% | 0.0% |
By turn
| Turn | On the play | On the draw |
|---|
What are the odds of drawing a specific card in your opening hand?
In a 60 card deck with 4 copies of the card, you open it in your first 7 cards 39.9% of the time. With 3 copies it drops to 31.2%, and with 2 copies to 21.8%. On the draw you see one extra card, which takes 4 copies to 44.2%.
The reason a single copy of a 60 card deck is only 11.7% by turn 1 is that you are drawing without replacement: every card you have already seen is a card that can no longer be the one you want.
Why is the hypergeometric distribution used and not the binomial?
The binomial distribution assumes each draw is independent, which would be true if you shuffled the card back in every time. In a real game you do not: the deck shrinks with every draw and the odds of the next card change. The hypergeometric distribution is the one that describes drawing without replacement, and it is what this calculator uses.
The difference is not academic. Using the binomial on a 60 card deck with 4 copies and 7 cards drawn gives 38.1% instead of 39.9%, and the gap widens as you draw a larger share of the deck.
How many copies do I need to consistently draw a card?
As a rule of thumb in a 60 card deck: 4 copies gives you about 40% by your opening hand and about 60% by turn 5; 3 copies gives about 31% and 49%; 2 copies about 22% and 35%. If a card is essential to your gameplan, 4 copies plus a way to search for it is the only combination that reaches consistency.
In a 100 card Commander deck a single copy is only in your opening hand 7% of the time, which is why tutors matter so much in that format.
How does going first or second change my odds?
Being on the draw gives you one extra card, so every probability shifts by roughly one turn in your favour. In a 60 card deck with 4 copies, being on the draw by turn 3 is about the same as being on the play by turn 4. That extra card is the compensation for letting your opponent act first.
Does this calculator account for mulligans?
No. The numbers assume a single hand of the size you enter. A London mulligan lets you look at 7 new cards and then put cards on the bottom, which raises your chance of finding a specific card but costs you cards. To approximate two looks at a 7 card hand, take the chance of missing it once, square it, and subtract from one.
Reactions
0
0 Comments
Be the first to comment