Hey guys, thanks for clicking on my first article. I would like to tell you a quick story and then try to give you some powerful tools to work with when developing your stone deck and show you a real life example where math can help.

## The Set Up

It all began about a week ago when I was playing some side games with a local player. We were working best of three sets, really getting into the match ups and talking about the interactions and card choices: truly dissecting to gain understanding. One important note; he was playing a Burn list that splashed Darkness for Whisper from the Abyss and Flame of Outer World. I was playing a White Weenie deck that splashed red for all the usual suspects. As such our stone decks were acting in a very similar fashion. We were both playing a lot of Basic Stones, but we were supporting a second color as well.

In one of the games he flipped Magic Stone of Flame on turns one through four. This unfortunate event left him unable to cast any of his Darkness cards and ultimately set him back a lot as he had drawn at least a few Darkness cards at this point. He commented on this while we were playing and that sparked a discussion about how likely or unlikely it was for him to run into this situation. He felt it was very unlikely to “miss” Darkness Will four turns in a row. I wasn’t totally sold on the idea and felt he was being at least a little misguided by the Gambler’s Fallacy. (Quick math note: The gambler’s fallacy truly applies to independent events, like flipping a coin over and over. In this scenario, flipping Magic Stone of Flame on turn one does increase his odds of flipping Magic Stone of Scorched Bales on turn two, however I think a lot of players allow the idea behind the Gambler’s Fallacy to inflate how likely it becomes to get the Darkness Will after flipping one, or two or three Magic Stone of Flame.) After we talked it out a bit more, we realized we cannot really pursue the issue without knowing the numbers.

## The Math

When I got home I immediately loaded up my favorite **Hypergeometric Distribution Calculator**, found here, and got to work. I wanted to get the numbers in front of me and figure out if what happened the night before was an anomaly or just good old probability working it’s magic. If you load up the calculator, there are four fields you must fill to get the information we desire. First is the “Population size.” In this case, his population was his stone deck of ten cards, so I entered 10. Second we need to know the “Number of successes in population.” For our experiment a success is any stone that produces Darkness Will. His stone deck contained four copies of Magic Stone of Scorched Bales as his only Darkness Will, so I entered 4. Next is “Sample size.” This is where you enter the number of attempts. You can change this number to do different scenarios with the same stone deck, but for now I wanted to see how unlikely it was that he missed Darkness Will after flipping four stones. I typed in 4. Finally, the number we all care about “Number of successes in sample (x).” Here you enter how many of those successes from the second field you expect to see in the number of attempts from the third field. For example we wanted one of our four Magic Stone of Scorched Bales to appear in the top four stones of the deck. I entered 1 and clicked “Calculate” with anticipation.

## The Results

The results come out in five different ways. The line that I think matters most is the very bottom one. It says “Cumulative Probability: P(X __>__ 1).” This means what is the likelihood of getting at least one success, aka at least one Darkness Will. The number was fairly high: 92.8%. When you consider the complement, meaning the chance that he did not get Darkness Will, it would be about 7.2%. So, now we have the numbers. I don’t think his bewilderment was totally unfounded. Being unsuccessful when he was 92.8% likely to be successful seems heartbreaking. However, consider that chance played out over and over and over again. We played ten games that night and the relatively unlikely event only happened once. Meaning he was successful 9/10 times in flipping Darkness Will on or before turn four. He was operating at 90%. If we compare 90% to 92.8%, I’d say everything was going according to plan and if we keep playing and keep playing and keep playing his success rate should climb a few points and settle in right around 92.8%.

So lets take a quick look at my stone deck. I was playing one card for my splash that I think makes all the difference. I had access to Fire Will via four Magic Stone of Heat Ray and one copy of Magic Stone of Moon Shade. Now, lets recreate the experiment above, but this time I enter 5 into the field labeled “Number of successes in population.” Click Calculate and take a look at that last line of results. With five sources of Red Will, I am 97.6% likely to draw one by turn four. The compliment being 2.4% basically means I will only be unsuccessful about 2 or 3 games out of 100. Look closely at the rate we are both unsuccessful; 7.2% compared to 2.4%. With our respective stone deck configurations, he is going to be unsuccessful three times as often as I am. Over the course of an entire ARG event or a Regional Qualifier that could really hurt.

## Considerations

An interesting thing I noticed when looking over the ARG St. Louis Top 8 decklists was the relative lack of Stoning to Death. The Ebony Prophet/Abdul Alhazred, the Harbinger of Despair decks had a total of two copies of Stoning to Death, and those were in the sideboard. The Liberator of Wind/Scheherazade, the Teller of 1001 Stories decks had a total of six copies, all main deck. I can attribute this to the difficulty of reliably getting double Darkness Will in a timely manner in the Abdul decks, while the Scheherazade players are fixing their stone deck before the game and increasing their chances drastically. With what we’ve explored above, you can use the calculator and determine just how likely it is for each of these decks to be successful casting a double Darkness card on turn two, or three or beyond.

As a very important final note, these numbers do not tell the whole story. There is a lot of outside information to consider before just jamming a Magic Stone of Moon Shade into every deck you play. Some important points might include: Can my deck survive if I have to take 200 damage a couple of times to produce will? Is my deck very vulnerable to Split Heaven and Earth such that increasing my special stone count is too dangerous? What one stone change could I make? Magic Stone of Darkness, Grusbalesta, the Sealing Stone, Magic Stone of Moon Shade all provide different advantages and disadvantages and any one of them would help my opponent more consistently flip Darkness Will.

Thanks for reading.

~Ken

There’s an interesting article by C. Mascioli about MtG and the hypergeometric function, available here

http://www.gatheringmagic.com/chrismascioli-100512-of-math-and-magic-part-1-the-hypergeometric-distribution/

Also, FoWDB implements the “Calc” function that has pretty much the same form as stattrek but in a more TCG oriented feel, here

http://www.fowdb.altervista.org/?p=calc

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This is awesome keep it up !!!

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