[Tool] A card draw probability analyzer

Summary: No detail statistics here, but simply providing a table and a excel sheet to automatically compute probability of drawing at least one card in your hand by X draw.

Probability Analyzer

What is this for?

To answer the question of what’s my chance of having a specific card in my hand after X draw.

How to use this?

The row represents # of cards you have drawn so far i.e. at the beginning of game if you go first, you have 6 cards.  If you go second you have 7 cards.  Each turn you draw additional card, simply move down the row.  If you use a drawing engine and draw more then you basically get down further earlier turns.  The column represents number of copies you have in your deck i.e. 1 to 4 in Shadow Era.

So example, if you want to draw Aldon by third turn which is when you can really cast him if you go first you will have at least 8 cards drawn.  So you are at row 8.  If you have 4 copies in your deck you have 61.75% (a little less than 2 in 3) chance of having him in your hand but if you have only 1 copy, you only have 20.51% (1 in 5).

Probability Analyzer Sheet

The above table was simply for the quick reference. If you want to see up to full 40 cards (39 technically as Hero is excluded) or more importantly, if you want to do calculation for more than 4 copies Open Probability Analyzer Sheet.

The use of above sheet is simple, you only can entire two variables.

  1. Deck size – How many cards your deck really has including Hero.  *The hero card not being part of draw is included in part of calculation so you don’t need to exclude in this value.
  2. # of copies in deck – How many copies of the target card you have in your deck.  E.g. single card ranges from 1-4, which are already in table; however, if you want to do more something like I need any ally with cost of 1 and you have 8 of those, simply enter 8 here.

Real world use

You can answer several things from this.

70% rule

First to answer to be able to utilize this probability table, you need to have set goal probability i.e. acceptable probability that you say if I have this much chance of having this card in my hand, I am satisfied.  Given the good winning rate of most TCG is something like >70-80%, if you define the card as your key card and having it to win the game is the goal (which most cases I assume), use this number as cut off.  So let’s say 70%.

In 40 cards deck, 70% target can be achieved in following number of turns using the analyzer tool.

Specific Examples

How many copies should I include?

Based on the analyzer tool, if you only have a single copy of a card, 70% goal won’t happen until 28th draw. Two copies 18th draw.  Three copies 13th draw.  Four copies 10th draw.  Draws can easily be converted to minimum turns by counting rows from top or simply subtract starting draw and add 1. i.e. 28-6 = 23 turns. 10 – 6 = 5th turn if you go first.  If you go second use 7 to subtract instead.  So this basically says if you have four copies you have a copy of card you need by 5th turn 70% of time.  So card like King’s Price that cost 7 to cast, almost no point of including 4 copies in 40 cards deck as you can’t even cast the card during first 6th turnss

I need to have any ally card with cost 1 that I can cast on my first turn if I go second.

Very specific question here.  But since you are stating you need this card when you go second, you are already looking at second row (# of cards draw = 7).  Even with 4 copies, you don’t have > 70% goal.  So go to Probability Analyzer Sheet.  Change # of copies and look at N column.  You will find out after entering number 6, you get >70%.  So you need at 6 allies costing 1 to meet this target.

40 vs. 50 cards deck?

Again for this go to Probability Analyzer Sheet.   Change deck size and see what exactly happens.

If you are going to make 50 cards deck, even if you include 4 of a kind in your deck, the earliest you can achieve 70% target is 8th or 7th turns. This means that you can essentially achieve exactly same probability by cutting down cards to 3 copies in 40 cards deck. In theory, you can cut 10 sets of 4 copy cards in 40 cards deck down to 3 each, you open up extra 10 cards space yet achieving exactly same probability potential as the 50 cards. In fact, one benefit of keeping to 40 cards is that you can achieve 70% goal earlier than 8th or 7th turn but once you have 50 cards deck you can never do so.  Though this may not be the sole reason, perhaps this is the reason why all the top players’ deck example are 40 cards deck.

More Advanced question

How about Y copies by X turn?

For the general discussion, I refer you to http://www.shadowera.com/showthread.php?2077-The-easy-way-to-calc-probabilities-for-your-deck-designs.  Here you can essentially do calculation such as I want to have multiple (2, 3, 4 etc.) copies in my hand.

Note: The OR analysis part has I believe flaw in the statement as you cannot simply add and divide two probabilities to get real number, but I believe instead you have to do add and subtract AND probability.

But essentially you can open this calculator. Then plug in the following values:

  • Population size: Your deck size minus 1.  Here you have to account for hero on your own as this is generic calculator.
  • Sample size: How many cards you’ve drawn so far.  Again, for the players going first, this value start outs with 6 and going second starts with 7.
  • Number of successes in the population: How many copies of this card you have in your deck.
  • Number of successes in sample: How many copies of this card you need in your hand?  If you enter 1 here, you can essentially use my calculation sheet above for simplicity but if you want to do 2 or more, use this approach.

Source: http://www.shadowera.com/showthread.php?2077-The-easy-way-to-calc-probabilities-for-your-deck-designs

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