In Magic, at the start of the game, you draw 7 cards. How would you calculate the likelihood of drawing a specific card in your opening hand?
For example, let's say I have a 60 card deck, and I'm running 4 Birds of Paradise. What is the percent chance that I will have at least one Bird in my opening hand?
The calculation you are looking for is called a Hypergeometric Distribution. This calculated your chances of drawing a particular number of "successes" from a population, without replacement.
Successes in Sample: 1 (the minimum number we want to draw)
Results: 39% chance of drawing at least 1 Birds of Paradise.
In the Hypergeometric Distribution calculator linked above, that result is represented in the Cumulative Probability: P(X ≥ 1) field: the chance of drawing greater than or equal to 1.
The online calculator will also give you the odds of drawing greater than that many successes in the sample (6%, the P(X > 1) result), and exactly that number (33%, the Hypergeometric Probability: P(X = 1) result).
You can see the calculation on the Wikipedia page, or searching math.stackexchange.com for Hypergeometric Distribution. Unfortunately, this site doesn't support math formatting. (Note: You will also need to know how to calculate binomial coefficients (and factorials).
The odds of drawing a particular card in a 60-card deck are obviously 1/60. If there are four such cards, the odds are 4/60. The odds of NOT drawing one of those cards in the first draw is 1 - 4/60 = 56/60.
To calculate the odds of the entire first hand, we can do it backwards:
The odds of not having any of the four cards in the first card is 56/60 (as I said above). The second card has odds of 55/59 (i.e. one of the remaining non-Bird cards after a non-Bird card was drawn to start), and then 54/58 and so on:
The odds of ALL of these happening (i.e. none of the four cards being in your hand) is the result of multiplying all these odds together:
(56*55*54*53*52*51*50) / (60*59*58*57*56*55*54) = ~0.6005 or ~60%
To calculate the odds of at least one of these cards being the one you're looking for, you can subtract this result from 1 (or 100%) to get a ~40% chance that (at least) one of your four cards will occur in a 7-card draw from a 60-card deck.
Magic Workstation besides many other tools for collection management, deck building, and online play has a very powerful probability calculator. It will go beyond opening hand and will let you see by what turn are you likely to have drawn the combo that you need.
