When building decks and evaluating how consistently they?ll play, many people struggle with how to calculate the probability of:
- Getting at least one specific card A by the turn they need it to appear, while balancing that probability against having too many of that card when they don?t want it to appear.
- Getting a combo of card A AND card B, either by the same turn or card A on turn X and card B on turn Y.
- Getting EITHER card A OR card B by a specific turn.
- Extrapolating from specific cards to general events, the odds of both event A AND event B happening, or of EITHER event A OR event B happening.
1 ? How to calculate the probability for any number of the same card on a specific turn
Let?s tackle the first bullet, because it?s the basis for how to answer all the other bullets.
There is one ?best-fit? formula for finding the basic probability inherent in all these types of questions that revolve around card decks. It?s called a hypergeometric distribution. Excel has a function for this type of distribution, but it?s even easier in many cases to use a free web-based calculator. Here?s the best one I?ve found. It also has decent links to educational/tutorial material on probability theory too.
Start here: http://stattrek.com/Tables/Hypergeometric.aspx
Then plug in the following values:
A ? Population size: Your deck size minus 1 (to account for your hero card). For example, if you have a 31-card deck, this value must be 30. If you have a 35-card deck, this value should be 34, etc.
B ? Sample size: How many cards you?ve drawn so far by a given turn. For players with the first turn, this value must be: 6 for turn one, 7 for turn two, 8 for turn three, etc. For players with the second turn, this value must be: 7 for turn one, 8 for turn two, 9 for turn three, etc.
C ? Number of successes in the population: How many copies of this card you have in your deck. For example, if you?re checking probability for pulling two Sandras by turn five, and you have 4 Sandras in your deck, this value must be 4.
D ? Number of successes in sample: How many copies of this card are you testing for? For example, if you?re checking probability for pulling two Sandras by turn five, this value must be 2.
You can play with these numbers all you like:
- Change the value of A to test for different deck sizes
- Change the value of C to test for different amounts of each card in your deck. For example, 2 Aeons versus 3 Aeons.
You folks looking for a leg up on building a high-probability deck: there ya go.. this is the golden key that all the tourney MtG players use to build their decks.
2 ? How to calculate AND probabililties
Per the second bullet above, it?s very easy to calculate the probability of card A AND card B, either on the same turn or different turns.
First, understand that this is different from the probability of having two or three of the *same* card by a given turn. The calculator itself can tell you that by simply using 2 or 3 for the D value.
But what if you want to know the probability for drawing something like ?at least one copy of card X on turn 2? AND ?at least one copy of card Y on turn 4?? To figure this, you simply multiply the probability for one event against the probability for the other event. The result is your probability for BOTH events occurring.
(Px)(Py) = PxANDy
For example, use the calculator to get the probability for card X on turn 2, then do it again for card Y on turn 4, and then multiply those two probabilities together. The result is the probability that you?ll actually land card X on turn 2 AND ALSO land card Y on turn 4.
3 ? How to calculate OR probabilities
Per the 3rd bullet above, it?s very easy to calculate the probability of card A OR card B, either on the same turn or different turns. In this case, you simply ADD the two probabilities together, then divide by 2.
(Px + Py) / 2 = PxORy
4 ? How to calculate probabilities for any combination of AND and OR scenarios
Just throwing this in for good measure. You can use the same basic formulas for AND or OR probabilities to figure the odds of any number of events occurring in any combination of AND or OR clauses.
For example, you want to know the probability of having two Sparks on the board by turn 2 or of having one Spark with an Extra Sharp on it by turn 2. Start by breaking this down into separate AND and OR clauses:
Clause A: (1x Spark on turn 1 AND 1x Spark on turn 2)
Clause B: (1x Spark on turn 1 AND 1x Extra Sharp on turn 2)
Now work out Clause A by figuring the odds for at least one Spark on turn 1, then for at least one Spark on turn 2 (and be sure to reduce the total number of Sparks in the deck to 2 for the second part, because you?re assuming you pulled one of them on turn 1). Now multiply those two values together per the rules for an AND probability. Now you have your probability for Clause A.
Work out Clause B in the same manner as for Clause A.
Now use the rules for an OR probability to put Clause A and Clause B together:
(Pa + Pb) / 2 = PaORb
Final note ? Beware temptation to make your decks larger than the minimum size allowed by the game rules
Beware the temptation to add just one or two ?extra cards? because they?re ?so good? and they seem to drop the probabilities you care about by only 3% or 4%. Top-tier decks usually run at exactly 30 cards for a reason. Ask any MMO player if equipping gear that provides ?only? an extra 3% or 4% critical hit chance is ?uber? or in end-game raiding or PvP. Top-level TCG players will also prize every extra percent chance they can muster for the critical turns of the game (which in this game are usually turns 2 and 4).
There?s also the issue of compounding this ?small? probability drop for any one card into a larger drop for a card combo that you want. If you really want a particular card to play on your turn 2 and another particular card to play on turn 4, then the resulting probability reduction for getting BOTH cards exactly when you want them will be even greater than the probability drop for just one of the cards. For example, what might be a ?mere? 4% drop for either one of the cards would compound to something more like a 6.5% drop for the chance of getting off the combo of BOTH cards.