This week’s Hammer of Math goes to the realm of fantasy football and looks at one of Kevin’s favorite games; Blood Bowl! We’ll take a look a the basics of this amazing game from Games Workshop and discuss the wonderful world of probability, risk management, and the fun that can be had in the newest edition.
Blood Bowl is a game of risk management; of carefully planning your actions in the hopes of ensuring success. At its core risk management is about two things; reducing the probability of failure and reducing the consequence should something actually fail. In Blood Bowl this means not only delaying the dreaded Turnover for as long as possible, but also ensuring that should a Turnover occur that your opponent will have as difficult a time succeeding on their turn as possible. In general Turnovers occur when a player ends up knocked over in some capacity, ends up off the pitch, or fails to maintain control of the ball. Sometimes this occurs as a result of the opposition, other times it’s your own fault.
Agility and Passing Tests
Agility and Passing tests in Blood Bowl are based on achieving a particular number with a d6 roll. A roll of 1 always fails, a roll of 6 always succeeds. The bad news is that this means that there’s always a chance that any action could fail. The good news is that, with a re-roll, no matter what you still might succeed 31% of the time. These probabilities apply to a huge number of probabilities including passing, catching, dodging, picking up the ball, receiving a handoff, and landing after a player is tossed.
Basic probability for characteristic tests.
Basic movement in the open is thankfully free of the probability of failure, making it and standing up one of the few things you can do with no risk of failure. This means that coaches should want to prioritize moving open players and standing prone models up first, as they will stay stationary or down if the there is a Turnover. The exception is if a player should choose to Rush, in which case they have to make a series of 2+ rolls to move 1, 2, or 3 (if they have the Sprint skill) extra squares. The math behind this is fairly obnoxious as success is sequential; if you fail two rolls in a row the number of re-rolls you have remaining is irrelevant. Note that the Sprint skill has a 87% chance of success for running three squares with a single re-roll, plus you have the advantage of being able to stop at any time.
The act of passing is amusingly complicated in Blood Bowl, requiring at least two skill rolls and the possibility of an intermediate player intercepting the ball. The skill checks use the probability from the attribute test section, but what’s particularly interesting is that if the pass fails then there’s still a chance that the receiving player could get the ball. When the ball scatters it randomly jumps one of eight possible directions a total of three times, starting at the intended landing point. Because the first shift occurs from the center this provides a very interesting result; you actually have a better chance of catching a scattered ball if you’re one square away from the target destination. This also makes the Diving Catch skill extremely useful, as it means a player has a 47% chance of being able to catch an inaccurate pass (and a 17% chance of being able to catch a wildly inaccurate pass, which deviates 1d6 squares from the target).
The core of any game of Blood Bowl, blocking can easily be one of the riskiest (and most rewarding) moves a player makes. The blocking action requires that you either be adjacent to a target or initiate a blitz to move next to them. At that point you roll a number of blocking dice depending on the difference in Strength characteristics of you and your opponent. There are five possible outcomes on a block; player down, both down, push back, stumble, and pow! The block die has two faces with push back and one face with all of the others, meaning that before modifications every die roll has a 33% chance of coming up with a result that could cause a Turnover.
The difference in Strength characteristic determines both the number of blocking dice and who gets to pick the result. The Strength characteristics of both players can be modified by assists and skills. If both players have the same Strength characteristic then a single die is rolled and the result applied. If one player is stronger than the other then two dice are rolled and the stronger player chooses. If one player has a Strength characteristic that’s more than double the opponent then three block dice are rolled and the stronger player chooses the result. Properly positioning players to get good blocks is crucial.
The chart below shows the probability of getting at least one result for each die roll. Depending on who is stronger will determine which result is preferable. For example if you want a
Note the symmetry of the potential outcomes. Every roll has an equal chance of knocking down just the player (player down) or just the target (pow result). Against an opponent with the Dodge skill fully half the possible outcomes could be considered neutral (non-damaging) outcomes, while if the player doesn’t have Block the chance of an outcome that could result in a Turnover (player down or both down) is equal to the chance of a push back.
Of course there’s another outcome that players are fully aware of; triple skulls. Even the best possible situation where you’re throwing a three die block has a 0.5% chance of guaranteeing three player down results, and without the Block skill that chance jumps up to 3.7% since both a player down and both down result will force a turnover.
Armour and Injury Rolls
Armour Rolls consist of rolling a 2d6; if the result is equal to or higher than the target value then the armour is broken and an Injury roll is made. Because the roll is made on a 2d6 the probability increases significantly the closer you get to the mean value of 7; going from a 10+ to a 9+ is only an 11% increase while going from an 8+ to a 7+ is a 16% increase. The probabilities for getting a particular injury result are also based on a 2d6 roll. Each roll has a 17% chance of causing a casualty, a 25% chance of causing a KO, and a 58% chance of stunning the target.
Of course there’s more than one way to injure a player. You could block them, but if they’re already down why not get some friends and foul them? Fouling is an excellent way to increase the chance of breaking armour as every assist will affect the Armour roll and the more a roll is modified the greater the cumulative impact. Because a Foul can cause an ejection on a doubles the probability of it occurring is a fixed 17%. If the coach wants to Argue the Call the cumulative probability of being Sent-off every time a Foul is made is 14%.
As you can see there’s a lot behind the rolls of a Blood Bowl game, and understanding the interactions and probabilities is a key part of ensuring you minimize risk and the chance of a Turnover. Making informed decisions, understanding the consequences of play, choosing skills which maximize your chance of success, and choosing the order of actions are all essential for good play. Of course the knowledge that the probability of rolling a triple skull is only half a percent is cold comfort when it occurs on a crucial blitz that would have stopped the winning score, but that’s just Blood Bowl.
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