Probability, part 2
Okay, so all this theory is fun, but how can it make you a better Magic player? Sure, the last post helps with deckbuilding – a little – but most of those numbers have been really well explored by now, and vary more based on other factors than math.
So let’s try and tackle something a little more concrete: The decision on when, and when not, to mulligan.
I suspect this topic might span more than one post, so bear with me.
Whenever you’re tackling a problem like this, it’s good to define your terms. We all know what a mulligan is, it’s discarding your starting hand based on the information you gain from looking at it in the hopes of a better one. What does “better than ” mean? Well, it can mean any number of things, depending on your deck. It could mean you’re likely to draw your combo pieces, or maybe you want to maximize copies of a certain card. But in general, I think that a successful opening hand means that, by the time the game gets underway, you’ll both draw enough lands to cast your spells, and enough actual spells to cast. Again, for each deck this is going to be different; fast decks are going to want more spells, more control-ish or combo decks might need more consistent land drops.
I’m going to introduce a bit of a new concept here, namely the random variable. Which doesn’t mean that you pick a letter at random instead of x, it’s actually a very specific kind of function that has to do with assigning a certain event to a number. For example, the number of lands in your opening hand is a random variable, call it X. You can represent the probability of certain numbers of lands in your opening hand by P(X=3), for example.
A good metric to measure the quality of an opening hand is how long it takes to be reasonably sure of drawing a certain number of lands, and the same for spells. For a baseline, let’s look at a deck with 35 spells and 25 lands. We’ll define our random variable L[n] as the number of turns it takes to draw n lands and, along the same lines, S[n] as the number of turns it takes to draw n non-land spells. Remember that random variables are a number that are associated with a certain probability, in the sense that we are interested in things like what P(L=3) is.
Actually, let’s go ahead and calculate that. Using our formula from last week, we have
assuming we’re playing first which, for the sake of argument, I’ll be assuming here throughout this series of articles. As a reminder, C(a,b) is the binomial coefficient, sometimes also called “a choose b”.
What is interesting for us to help us decide when to mulligan is the average number of turns we’ll need to draw a given number of lands – we’ll say, for the sake of argument, 4, since most decks want to make their first four land drops pretty consistently. When talking about random variables, this “average” is called the expectancy.
When you calculate the average of a set of numbers, for example, you add up the numbers, and then divide by n, the number of numbers in the set. The expectancy is very similar, if you look at the numbers in the set earlier all to have the same probability of occurring, namely 1/n.
When we calculate the expectancy, we again add up all the numbers that can occur in the random variable, but we multiply each one first by the probability it has of occurring. So, in our example above, we take the number of turns “3”, and multiply it by the probability that by turn 3 we have drawn exactly 3 lands, which is 22,9% (-ish). we do this for all conceivable numbers, which would be infinitely many, but luckily we can pretty much stop counting after 60, since by that point you’d have drawn all of your deck anyway. The formula for this is written:
E(L[n]) = Σ (P(L[n]=X) * X)
and the result is simply a number, in our case one that lies somewhere between 0 and 60. As a reminder, Σ is used to denote a summation, in other words we add up all the terms that appear after it. I left off the range because it should be clear that we’re adding over all integers, although in our case we can stop after 53. Of course, actually calculating this can be a bit of a pain, but there are a few shortcuts.
I’ll delve into those, and how we can use the expectancy to help our Mulliganing decisions, next week.
- Given a deck with a 35/25 land ratio, what is the probability that, on turn four, you will have drawn exactly four lands? What about at least four lands?
- How are the random variables L[n] and S[m] connected? Can you make a formula to calculate one based on the other?
- What is the expetency for the random variable of “number of lands drawn in your opening hand” for a deck with a 35/25 land ratio?