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If I went through the audience one by one and got each person to call out the date and month they were born, how many people would I have to go through on average, before the same birth date is called out twice?

Some people called out numbers which were probably random. A couple of people called out 182, which was my guess - that was halfway through the year, you'd have to be at 50%.

The answer is 23.

Let's start talking about Magic, I'll put the working out for that question at the end if anyone wants to work their way through it. I enjoyed maths at school, and the problem solving nature. I hope this to be the first in a series of articles on probability in Magic, starting simple and getting more complex as I re-explore (and re-educate myself in) this interesting field of maths. Perhaps these articles won't be for everybody, but I think they will help me to make better deck building choices, and hopefully the information will help you too.

So we will start simple, and talk about how much land you should put in a deck. When I started, you got a starter deck with 20 lands in it, so we always thought you needed about 20 in a 60 card deck. Nowadays, we realise that 24 is about the right number. Why? Well, playing experience tells us that it lets us get enough land to cast our spells a bit earlier. Let's look at the maths behind it.

Say in a 60 card deck, you have 20 lands (one third, or 33.3%). If you draw one card from the top of that deck, there is a 33.3% chance that it will be land. But note now that if you do, the probability of the next card being a land has changed; it is no longer 33.3%. There are now 19 lands left in a 59 card deck. Given that the first card drawn is already a land, the chance of the second being a land is 32.2%. Here is where it gets tricky - what is the chance of drawing 2 lands in a row off the top of this 60 card deck? The answer is 10.7%. It is the chance of the first card being a land, multiplied by the chance of the second card then being a land (33.3% x 32.2%). For completeness, the chance of the 2 cards being non-land cards is 66.6% (40/60) times by 66.1% (39/59) which is 44.1%. The chance of drawing a land and then nonland is 33.3% multiplied by 67.8% (40 nonland cards left out of 59 cards) which is 22.6%. The chance of drawing a nonland card then a land is 66.6% multipled by 33.9% which is 22.6% (noting that this is the same as drawing a land then nonland).

That was just to start us off on learning the numbers. Of course, when we start playing, we already have 7 cards in our hand. Each deck will have different mana requirements, and it is good to understand how soon we need that mana to make our deck work, and how many lands we need to make sure we get there in ample time. So when you draw those initial 7 cards, what are you chances of getting land if we use the 20 land deck?

Using a Hypergeometric distribution table from Excel, the chances of drawing X number of land from your first 7 cards in a 20 land, 60 card deck are:

This means that 57.1% of the time, you will draw 7 cards with 2 or less lands. Almost 1 in 4 games you will have 0 or 1 land in an opening 7 card hand.

So how is this information useful? Realistically, this alone is not that useful. Unless you play a deck with spells that all cost one mana, your deck optimally will want to hit a land drop each turn for a certain number of turns to make sure it can start casting spells quickly and effectively. Say for example that on turn 3 you want to make sure that you hit your 3rd land drop. Assuming you are on the play, after your 3rd turn draw phase you will have drawn 9 cards. Analysis shows that after 9 draws, the chance of you having drawn X lands are:

This means that on your 3rd turn, 36.2% of the time you will be stuck with 2 or less lands. That's slightly more than one in every 3 games, which personally is not consistent enough for me, or for most people if they are looking to play a lot of spells that cost 3 mana or more. So lets look at the 'norm', 24 lands, and compare with 20, at 7 cards, and at 9 cards.

Running 24 land means you have a 41.2% chance of getting 2 or less lands in your opening hand, compared to 57.1% if you run 20. More importantly, 24 lands mean that on the 3rd turn, we will fail to make our 3rd land drop 21.1% of the time, compared to 36.2% with 20 lands. 1 out of every 5 games is a lot better than 1 in 3. Of course, this works in reverse as well, as the more land we run, the more chance we have of mana flood. In a 20 land deck, there is only a 12.6% chance we will have at least 2 land still in hand after the 3rd turn land drop. In the 24 land deck, there is a 25.1 % chance that we will still have at least 2 lands in hand. Whether this is good or bad is deck dependant. If all spells in the deck cost 3 to cast, perhaps 24 lands is too many, and you can afford to drop 1 or 2.

We'll take a look at a casual play deck idea that I have had in the back of my mind for a while, and apply this formula to it. This allows us to look at the deck without having to try and build it or proxy it up, then playtest it to find out the land ratio is incorrect. (Playtesting is nearly always necessary, this just gets the first build closer to ideal and can save time).

My initial deck idea was:- How soon can a deck go off that focuses on getting out a plethora of greens mana creatures for a single fireball to my opponents head?"

After the idea sat in the back of my head for a while, I added three other non-creature cards (apart from Fireball); Collective Unconscious, Glimpse of Nature, and Ashnod's Altar; for card drawing and allowing the 'combo' (per se) to go off sooner. If I used all of green's cheapest mana producing creatures, how many lands should I use? Obviously for the rest of the deck I wanted to include as high a creature:land ratio as I could handle, as the more creatures in the deck, the better all of the non-creature cards would work. With Llanowar Elves, Birds of Paradise and Fyndhorn Elves, I figured the deck could stand to start with only one land in hand, but it was preferable to get another land out on second turn to power up a Glimpse of Nature, or cast more creatures to power up a Collective Unconscious, a little bit sooner. Let's have a look at the ratios for different numbers of lands at 8 cards (second turn on the play), 7 cards, and 6 cards:

The first table of 8 cards helps when designing the deck to see how often you will get 2 lands by the second turn, assuming you are on the play. The second table of 7 cards shows you your opening hand. The third table of cards is to show your chances if you mulligan your opening hand down to 6. This is useful, because we can determine that if we have a poor opening hand, we can then see if our chances are better if we mulligan. For example, say we run 10 lands, there is a 25.9% chance that an opening hand of 7 will have no lands. If we mulligan, our chance of getting no land out of 6 cards is 31.7%. This makes for an 8.2% (25.9% of the first thing happening, times the 31.7% chance of the second happening) chance that at the start of a game, we will be left with no lands after mulliganing down to 6. We can keep going of course, but we start heading into serious card disadvantage.

Based on the above, I would start testing the deck with 13 lands. At 13 lands, there is about a 1 in 6 chance of an opening hand containing no lands. When they do and I mulligan down to 6 cards, there is about a 1 in 5 chance that it will happen again, and I am comfortable with those odds as a starting point.

Hopefully some of these points have helped the way you look at deck building. Most of the decisions that probability can help you with, you already make intuitively. Perhaps the information provided here can help you make them with more precision. An Excel spreadsheet that has been designed to easily modify the numbers and update the percentages is available; if you would like a copy, please e-mail me and I will send it to you.

Some assumptions during this article:

• The ideas presented here assume you are on the play. The odds of drawing land by a certain turn improve if you draw first, but your deck must be able to handle playing first.
• When determining the probability of lands by a certain turn, it does not take into account any deck-thinning or card drawing. (The deck idea presented above would benefit strongly from Onslaught fetch lands, but is beyond the scope of this article)
• The tables are for land in general. For example, even if a deck has mainly low cost spells, it might need to have more land if it is 3 colours.
• The article assumes the deck is completely random. Picking up a bunch of lands after a game and shuffling only a few times may leave 'land pockets' in the deck. People may also have a pattern to their shuffling which does not make the deck completely random.

What I would like to cover in future articles:

• The usefulness of deck thinning and land search.
• Probabilities involved in getting combo cards in hand.
• Serum Powder, Fiery Gambit, Krark's Thumb and other oddities whose use relates to probability.
• Probabilities of getting the land you need in multi-colour decks.
• Other issues you readers might like to hear about (so long as I can get my head around them).

Your feedback is greatly appreciated so that future articles I write meet your expectations.

The answer to the opening question:-
"If I went through the audience one by one and got each person to call out the date and month they were born, how many people would I have to go through on average, before the same birth date is called out twice?"

The first person calls out a birth date. When the second person calls out a birthdate, there is only one date they can call out that will be a match. Therefore, there is a 1/365 chance that it will be the same. The third person will have 2 days that can match, meaning 2/365 times by the chance that it has not already been matched.

To get the answer, you need to work backwards for this one, and find the chance that it doesn't match until you meet the average, 50%. So the answer is (364/365) for the second person*(363/365)*(362/365)... until you get to the 23rd person (343/365), at which point the answer becomes very close to 50%, the average.

Formula

And the formula for working out the probabilities is:

a = no of lands in deck
b = no of non-lands in deck
c = no of cards in deck
d = no of cards to draw
n = the required number you are testing the probability of

For those unaware, the '!' is the factorial operator. Factorial means that number, times by each integer below it down to 1. (e.g. 5! = 5*4*3*2*1 = 120)

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