Much has been said about how the game of Quidditch is ruined by the scoring system – specifically how it makes no sense that the snitch is worth 150 points and that catching it ends the game [1, 2, 3]. Most of these arguments seem to revolve around the claim that it is nearly impossible to win a match of Quidditch without catching the snitch. Is this true? Let’s try and answer this question formally using statistics and R simulations:
What is the probability *of winning a Quidditch match without catching the snitch? |
| |Victor Krum caught the snitch for Bulgaria, but Ireland still won (1994 World cup)*|
The Game, the Rules
Quidditch is a fictional team sport played on broomsticks, devised by author J. K. Rowling, that is played by wizards in the fantasy world of Harry Potter. Like any sport, the game of Quidditch has many rules, but I will summarize here only the ones pertinent to our discussion:
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10 points are gained by throwing the Quaffle through one of the opponent team’s three hoops.
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150 points are gained by catching the Golden Snitch, which can only be caught by the Seeker.
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The game ends when the Golden Snitch is caught.
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The winning team is the one has the most points at the end of the game.
Given these rules and game objectives, in order to estimate the probability of winning a match we must first estimate (1) how long does it take a seeker to catch the Snitch on average, which determines how long a game lasts and thus how much time is allotted for the teams to score goals, and (2) what is the rate of goal scoring.
Estimating the Average Game Length
For the sake of our simulations, then, I will assume the average game lasts 90 minutes (assuming Rowling has soccer or rugby in mind). I will also assume that the length of a Quidditch game is exponentially distributed, since the probability of catching the Snitch remains constant (as opposed to it getting easier as the game progresses) – the Snitch’s ability to escape capture does not change (magic) and players are substituted throughout the game, so let’s assume an unchanging level of vigilance.
Together, the distribution of game lengths looks something like this:
This makes the 6 month long game very unlikely ( $Pr(time>260,000)<10^{-10,000,000}$ ), but the 15 minute game rather probable ( $Pr(time<15)=0.15$ ).
Estimating the Rate of Goals Scored
I will also assume that this rate is constant and unchanging throughout the game, regardless of the score (or fatigue of the players, see above), and so will assume GPM has a poisson distribution, with an average of 0.8, making the GPM distribution look something like this:
Simulating the Games
(If you’re not that into reading raw code, you can just skip to the conclusions at the end.)
R Setup
Equal Teams Let’s start by simulating 1 million games between two teams with equal GPMs and equally good seekers.