Hypergeometric Distributions and You

A hypergeometric distribution is a decimal that represents the likelihood that W successes will occur within a given sample given that there are X 'chances' within the sample (the sample size), Y instances of the successful event in the population as a whole, and Z events in the population.

In short, it is the odds of a person having a certain card in hand.  The information from a hypergeometric distribution can be combined with the the information the other player is giving you to help you to determine whether or not he has the card in question (or a similar card).

Let us suppose, for instance, that you are playing an aggressive deck against a more controlling deck.  The game is going a little later than you would like... You have reached the 8th turn.  Your opponent loves Day of Judgment, so you can be certain his deck contains at least 3 if not 4 of them.  He has not played any yet this game. After burning through his chump blockers, you are starting to get in for decent damage.  You have a pair of Goblin Guides and a Signal Pest on the field currently.  With your opponent at 8 life, you are starting to gain confidence.  Unfortunately, you have good reason to fear a Day of Judgment.  With your hand clogged with cheap creatures and lands, do you commit more creatures to the battlefield or wait until after the Day of Judgment to continue your assault?  Time for some number crunching.  Eighth turn.  Your opponent was on the draw, so 7+7+2=16 is the number of cards that have been through his hand this game.  What is the 2 for, you may ask?  He played Divination, so those cards must be taken into account.  This gives me the formula: =hypgeomdist(1;16;4;60) [Hint: replace the semicolons with commas if using Excel; I use Google Docs].  The 1 is the number of the particular card in question (day of judgment) we are interested in him/her having.  The 16 is the number of cards the player has seen this turn.  The 4 is the number of the card (or cards) of interest in the deck.  The 60 is the size of the deck.  The result is about 0.43, or 43%.  This is a reasonably high pobability.  Now the question is, what do you think your chances of winning are after he lands a Day of Judgment?  If the odds are higher given the hasty creatures you have in hand, your best bet would be to wait it out and get in for damage with the creatures already on the battlefield for the time being.  If you know your opponent packs some nasty creatures in his deck that will invariably hit the battlefield after he blows up the board, giving you almost zero chances of winning, you will need to press your advantage and hope that it is enough.

I will try to get a few charts up with some key sets of hypergeometric distributions on them for reference purposes.

If you would like to know more about hypergeometric distributions or you would like to see more posts like this, let me know by subscribing to my blog and leaving a comment.  Good luck out there.