Magic the Gathering Arena: Winning with Probability | by Edward Krueger | May, 2023


Many games and tournaments have complex rules. Why? They exploit our psychology to make the games more fun, addictive and often lucrative to the organizer by making our chances of winning appear higher than they are. It’s engaging when a fan, player or gambler believes they can win. This is most effective if you think your chances are better than they actually are.

In this article, we present a tournament from the MTG Arena game and calculate the odds of winning. When this tournament is discussed, its commonly thought that given even match-ups, a player will win 3 or 4 games on average, and a rough calculation will severely overestimate the odds of winning. We’ll do the math and show how to use Excel to calculate the odds — adapting the result to other statistical software should be straightforward.

Photo by Ryan Quintal on Unsplash

The Magic The Gathering Arena (MTGA) Decathlon consists of ten different events that take place over a month. Entry to an event is 2,000 gold or 400 gems. Events are either played as Best-Of-One (single-game matches) or Best-Of-Three (three-game matches). In this article, we discuss the expected outcomes relevant to a Best-Of-One (BO1) MTGA event. To earn a token, BO1 games require that a player wins 7 BO1 matches before incurring 3 losses. Given these parameters:

  1. What is the probability of earning a token when entering an event?
  2. What is the expected number of events a player must enter to win a token? Or, put another way, on average, how many events must a player enter to win 7 times, before losing 3 times, will it take to earn a token?

MTG is a game of both luck and skill. A highly skilled player will generally win more games, despite the variability introduced by luck. In the realm of mathematics, luck is quantified using probabilistic concepts.

Probability describes the uncertainty involved with a random process and is expressed as a number ranging from 0 to 1. A value of 0 indicates the event will not happen while a value of 1 indicates the event will happen. We describe a repeated random process in mathematical terms using the concept of the random variable, which represents the outputs of a repeated random process within a specified range. The probability distribution for a random variable is a statistical function describing how these outputs are distributed over the random variable.

Photo by Giorgio Trovato on Unsplash

Probability distributions map either discrete or continuous random variables from function inputs to real number outputs. Based on the parameters of our MTG problem, we know the Negative Binomial distribution models the random process inherent to our MTG problem. The Negative Binomial models repeated random processes with the following properties:

  • The process consists of repeated trials, each of which is independent, i.e. the outcome of each (x) trial does not affect the outcome of other trials.
  • The experiment will continue until some predetermined event (r) is observed. This predetermined event is often referred to as a “success”, however, it’s worth noting that it need not be a positive outcome!
  • Each trial has only two possible outcomes, an r success or an f failure, with the probability of success (p) being the same for each trial.

The Negative Binomial describes a discrete random variable and is therefore modeled by the type of probability distribution known as the Probability Mass Function or PMF, which specifically models discrete probability distributions. The PMF of the Negative Binomial, in which random variable X denotes the trial at which the rth success occurs (r-1), is defined as:

Equation by Authors

Each variable is defined as follows:

  • x: the number of trials to produce r successes in a negative binomial experiment
  • r: the number of “successes” in the negative binomial experiment
  • p: the probability of success on any given individual trial
  • q: the probability of failure on any given individual trial (equal to 1-p)

Alternatively, the negative binomial can be defined in terms of a random variable Y, which represents the number of failures that occur before the rth success (y). Note this alternative form to be statistically significant as Y = X — r. Transforming this yields Y + r = X.

Equation by Authors

So why does the PMF matter? It matters because it outputs the probability that a discrete random variable takes at a given value. To understand the probability that a discrete random variable takes at or less than a given value, we use the Cumulative Distribution Function or CDF, which is simply the probability that a random variable X is less than or equal to x.

The values taken on by the variables associated with the PMF of the Negative Binomial determine the exact shape of the Negative Binomial distribution. This shape is characterized by two (2) parameters, the stopping parameter, r, and the success probability, p. We, therefore, say the negative binomial has an (r,p) distribution.

Recall that we aim to model a problem where a player must win 7 BO1 matches before incurring 3 losses. In reality, a player can’t win more than 7 times, but to model this with the Negative Binomial distribution, we’ll allow 8, 9, 10, .. wins. This won’t affect our result because we’ll compute the probability of having 7 wins in the event as the probability of having more than 6 wins in the model.

What matters is the number of times the player loses. This is an important distinction to make as this determines our stopping parameter, r. The attempt is over when the player sustains 3 losses. Put another way, we are looking to calculate the probability that a player wins at least 7 times before incurring 3 losses. Because any number of wins greater than 7 would be acceptable, so long as the number of losses is less than 3, we would use the cumulative distribution function to model this problem.

Luckily, modeling problems like this is easier with computer software programs. Excel provides the NEGBINOM.DIST function, which requires the following inputs:

  • Number_f: the number of “failures”
  • Number_s: the number of “successes” -> the stopping parameter
  • Probability_s: the probability of a success
  • Cumulative: takes either TRUE, indicating a CDF problem, or FALSE, indicating a PMF problem

These arguments are set in the following order:

  • NEGBINOM.DIST(Number_f, Number_s, Probability_s, Cumulative)

We can define the probability of getting a token and the expected number of attempts to get a token at varying levels of game win rates (known as p within our Negative Binomial equation):

Figure by Authors

To get a token, a player must win at least 7 games before losing 3. So each event encompasses any number of games up to losing 3 games or winning 7.

So how did we calculate this? We calculated the probability of getting a token and the expected number of events to get a token by plugging the following into the NEGBINOM.DIST equation in Excel:

1 — NEGBINOM.DIST(Number_f=6, Number_s=3, Probability_s=1-game win rate, Cumulative=TRUE)

  • Number_f: Recall that the CDF expresses the probability that a random variable Y is less than or equal to the specified value, y, expressed mathematically as P(Y <= y). To find the probability of winning 7 games, we can find the probability of the complement: not winning 7 games first. Mathematically, this is P(Y < 7) = P(Y <= 6), which is the CDF evaluated at 6. Then we can calculate the probability of winning 7 games as P(Y=7) = 1 — P(Y<7) = 1 — P(Y<=6).
    Number_s: the stopping parameter, equal to 3 losses

We are defining “success” as a loss. And therefore, we set the second argument defining the number of “successes” or the stopping parameter to 3.

  • Probability_s: 1 — game win rate

Because we’ve defined a “success” as the probability that a player loses, we would use the inverse of the game win rate as the argument for the probability of success, equal to 1 — game win rate.

  • Cumulative: set to TRUE as this problem has been defined as a CDF

So to get a token with 50% of attempts, a player must have a game win rate of at least 71.36%.

A strong knowledge of probability will come in handy with any game involving chance. Here, we’ve shown how to apply probabilistic concepts to correctly calculate your odds of winning under different conditions. Understanding all possible outcomes, and the probability of occurrence of each outcome, facilitates more informed decision-making and hopefully, allows you to secure a win.

Sources:

  1. https://www.britannica.com/science/statistics/Random-variables-and-probability-distributions
  2. https://en.wikipedia.org/wiki/Random_variable



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