[{"text":"It\u2019s your resident BatCat (https:\/\/www.dcdeckbuilding.com\/article.php?id=10820) expert back at it again with another article. Unfortunately, this time I don\u2019t have a team to make bad MC\u2019s playable (Tempest is still super unplayable as a tier 1), but I hopefully have something more useful. You see, I\u2019m really good at math, and I\u2019ve been doing a lot of math lately based on certain aspects of the game. I\u2019d like to share some of my findings with you to hopefully help alleviate one of the most frustrating parts of the game: randomness."},{"header":"randomness"},{"text":"One of the biggest, most annoying sources of randomness in DCDB is the supervillain stack. We\u2019ve all had situations where a Supervillain swung the game either in our favor or against it, and it\u2019s fairly common for Supervillains to single handedly decide close games. First Appearance Attacks can devastate decks, and can affect certain players or teams more than others. These situations will continue to come up as there is no way to 100% prevent them, but there is a little bit you can do to help prepare for those nasty attacks. Before we get into the data, keep in mind that the costs I\u2019m using here are the base costs of the Supervillains. In competitive play, you need 2 more Power than the base cost in order to defeat the Supervillains. First let\u2019s look at the general distribution of supervillain costs and the probability of a certain cost appearing anywhere in the stack. The formula used here and all other formulas used in later tables will appear at the end of the article. "},{"image":[{"src":"cap960827capture.png"},{"caption":""}]},{"text":"So how can we use this table to our advantage? In practice, you will find this table by itself to not be very useful, but the information here is necessary to know in order to make sense of the later tables. Knowing that most of the Supervillain stack is made up of 10, 11, and 12 cost cards will make the probabilities shown later make a lot more sense. You can also use this table to come up with more useful things to know that I don\u2019t go over later in this article. Another piece of useful information to have would be the probabilities of certain costs appearing in certain positions. Let\u2019s say you are just starting a game and want to know the chance that the 2nd Supervillain costs 11 exactly. That\u2019s what this table will show."},{"image":[{"src":"cap986387capture.png"},{"caption":""}]},{"text":"This table is by far more useful than the first table. The probabilities in the last column effectively show the chances that you need no more than that much power to buy the whole supervillain stack. For example, in about 34.1% of your games, the final supervillain will be cost 13 or 14, so if your strategy is to buy the whole supervillain stack, you need to make sure you can get 15 or more power early enough. Otherwise, you might struggle in about \u2153 of your games. Similarly, if you are going a farming strategy and want games to go long, about 17.1% of your games will end very early because players will need only 13 power to buy the whole stack. It\u2019s worth noting that the latest banlist change made the supervillain stack slightly cheaper overall. "},{"header":"fun facts"},{"text":"You can also see a few surprising fun facts from this table:"},{"list":[{"listType":"bullet"},{"caption":"In about 1 in every 220 games, both currently legal 14 cost Supervillains will appear in the stack."},{"caption":"It is more likely that the final Supervillain will cost 11 than 14 even though 14 cost Supervillains would always appear in the final spot if they\u2019re in the stack."},{"caption":"While it is possible for every Supervillain in the stack to cost 9, the probability is so low that 38,508 games will need to be played in this format in order for the chances of it happening to be greater than 50%."}]},{"text":"Now lastly I want to show one more table. Unfortunately, due to size limitations I have to link to a Google Spreadsheat containing this table rather than including it in the article This table appears very convoluted, but I will try my best to explain its format and how to read it. This table is most useful during games as it shows the chances of the next supervillain\u2019s cost based on the position and cost of the current face-up Supervillain. The left column is the cost of the current face-up Supervillain and its position in the stack. The top row is the potential cost of the next Supervillain and the number of Supervillains that appeared that share the current face-up villains cost. https:\/\/docs.google.com\/spreadsheets\/d\/1h0x5Bl3LiJR164g0D8WhkxQJGqZnknoljl5wGq7zX6I\/edit?usp=sharing A quick guide to reading this chart. Let\u2019s say the current face up villain in the stack costs 11, is in the 3rd position, and 1 other cost 11 Supervillain has already appeared. Let\u2019s find the probability that the next Supervillain costs 12. To do that, we\u2019d look in the row labeled 11\/3 (cost 11 third position) and the column 12\/2 (finding cost 12 with 2 cost 11 Supervillains appearing before so far.) This leads us to cell 10J which shows 32.9%, so the chances of the next Supervillain costing 12 is 32.9%."},{"header":"how to use this information"},{"text":"Now it\u2019s worth noting that in official competitive games, you cannot bring any notes taken before the game, so in order to use any of the information in this article you\u2019d have to memorize it. However, I\u2019d recommend not memorizing all of it, especially the 3rd table as it has a lot of information to go through. However, it would be sufficient to know any extreme probabilities and general trends rather than every single one. For example, let\u2019s say the current face-up villain is in the 2nd position and costs 11. Now let\u2019s say during your turn, you reach 13 power and play Flight Wings. Should you stack the Supervillain to the top of your deck? Intuitively, you\u2019d think yes. Stacking it will guarantee you see it this deck cycle rather than the next deck cycle. However, if you know the general trends of the table, you\u2019d know that for 11+ cost villains that appear in the 2nd position, the next Supervillain is much more likely to be the same cost. You may also should know that there are some particularly brutal attacks in the 11 cost villain list. Most notably is Lex Luthor since his attack cannot be defended and will flood you with weaknesses if you have a high VP card in your hand. If you were to stack the Supervillain you defeat, and Lex Luthor flips over, you\u2019d gain weaknesses to cancel out the victory points you just earned. It\u2019s also worth noting that if you don\u2019t have defenses, Helspont IM, Mongul, Felix Faust, Brainiac, and Green Lantern\/Wonder Woman could be similarly negatively impacted by having a Supervillain in your hand. Meanwhile, none of the 11 cost Supervillains have attacks that get worse if a Supervillain is in your discard pile, and only Count Vertigo and Kobra care if a Supervillain is in your deck if you happen to be cycling your deck next turn. In this case, you might want to think twice about stacking the Supervillain to the top of your deck and assess the risk of doing so given the state of the game at the time.r"},{"header":"final thoughts"},{"text":"This was my first article about probability and DCDB, and I plan on writing more in the future. I want on the next one to be about the main deck, so if you have any suggestions, join the Discord server and shoot me a message. In the meantime, I hope the information in this article can help you win some games and alleviate the frustration of the Supervillain stack slightly."},{"image":[{"src":"cap83667capture.png"},{"caption":""}]},{"text":"Defining variables: a=number of Supervillains with the cost we are trying to find the probability of b=number of Supervillains with cost less than the one we are trying to find the probability of c=Position of Supervillain\/face-up supervillain d=number of Supervillains with current face-up cost e=number of Supervillains with cost less than current face-up cost f=1 if current face-up cost matches cost of villain we are finding the probability of, otherwise zero "}]