Mullligans, part 1

by MTGViolet

Hi Mathmagicians!

First off, some admin stuff. I’ve decided that, due to things like school starting up for me soon, I’m going to take this blog from weekly to biweekly. Which is honestly about as much magic as you might want anyway. On the off weeks I might post interesting links I find, but the articles will be posted once every other week from now on.

So, on to this week’s topic. When you should – and shouldn’t – mulligan.

Last week we learned about expectancies, and that’s the tool we’ll use for deciding when to throw back your hand or not. Now, if you just look at raw numbers, it seems clear that, if you have fewer cards in your starting hand, it’s going to take longer to draw your four or five lands that you want to make your chances of winning as high as possible. But doing so neglects an important aspect of making your mulligan decision, namely that you actually do get to see what’s in your opening hand. (If you didn’t, your mulligan decision would be truly easy – you just plain never would)

Let’s say you draw your opening hand and you see you have one land. Using the math from last week, the expected turn for you to draw your fourth land is:

E=1*Probability of drawing your fourth land on turn 1 + 2*Probability of drawing your fourth land on your second turn +…

What we are looking at here is a conditional expectation, (I keep saying expectancy, but the usual term is expectation. I learned all these terms in German, so sometimes things get lost in translation, sorry) which means that, just as it sounds, were taking the condition that we have a one-land opening hand into account when calculating our expectation. The formula for that is:

E[X|Y=y]=∑x*P(X=x|Y=y)

The | in the formula can be read as “under the condition that”, as in “the probability that you draw your fourth land on turn four under the condition that your opening hand has one land”. To calculate this, you take the probability that x and  y are both true, and divide that by how likely y is. In other words, you know that y is pretty unlikely to happen, so you eliminate that from the equation to see how unlikely x is to happen assuming y already has.

Got it? Don’t worry, it makes more sense when you actually do the math. Or at least I think so.

So, in our example, the probability of drawing one land in your opening hand is about 10.5%. The probability of drawing your fourth land on turn 4 after starting out with only one land is the same as saying each card draw is a land (by turn four, on the play, you draw exactly three cards) is 24/53*23/52*22/51, or about 8.6%, which is exactly the conditional probability P(X=4|Y=1). If we wanted to know the probability of drawing one land in our opening hand, followed by three more lands in our first three draws, we would multiply the two probabilities above to get 0.9%. That’s where the formula comes from.

To continue our reasoning above, it’s clear that the first three terms in our sum are zero (it’s impossible to draw three lands with two or fewer draws), so we start at turn four, which we calculated above. For turn five, if we want exactly four lands, we can either go Land Land Land Spell, or Land Land Spell Land, or Land Spell Land Land, or Spell Land Land Land, where the probabilities of all those events are the same. If this seems familiar at all, it should: It’s exactly the same math we used to determine the probability of  drawing a given number of lands in your opening hand, just adjusted for different variables. In this case, we have:

Number of cards in deck: 53 (we already drew our opening hand)
Number of Lands in deck: 24 (one is in our opening hand)
Number of cards drawn: Turn number – 1
Number of successes wanted: 3

Plugging those numbers into the formula gives us a probability function of

C(24,3)*C(29,t-4)/C(53,t-1)

and an expectation of

E[X|Y=1]=∑_(t=4..33)t*C(24,3)*C(29,t-4)/C(53,t-1)≈18.28

Yes… it’s that really high number again, but it’s okay, for the simple reason I stated above: The actual scale of that number isn’t as important as the fact that we HAVE a number.

Figuring out the conditional expectation of the various opening hands available is now a simple matter of plugging in a few different variables into a simple equation and pasting that into WolframAlpha. You can do that beforehand and write the numbers down.

The other nice thing about these numbers is we can use them in comparison to what might happen when you mulligan… but more on that next week… er, next fortnight.

Exercises:

  • Take your favorite deck. Calculate the conditional expectations for the different opening hands.
  • How can you calculate the analogous situation for spells versus lands.
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