In this era of rapid technological advancements, social media, algorithms, and artificial intelligence, everything moves at breakneck speed. Let’s take a breather. Let’s set aside our phones and recall our traditions: euskal dantzak, bertsolaritza, esku-pilota… We’ll take ourselves back to one of those slow summer afternoons, where, after a meal with friends, you play a game of mus. Time slows down.

Mus, briska or julepe are card games that are part of our culture, embedded in our lives for centuries. And yes, even here we can find math.

CHANCE AND PROBABILITY

Like any good game, a player’s skill is essential to win. But in card games, luck also plays a role. And here’s where things get interesting, as you’re left to play the hand you’re dealt.

Chance constantly disrupts our lives, and from experience, we can get a sense of the probability that certain random events will happen. Probability is the mathematical calculation that evaluates the likelihood of an event occurring when chance is involved. Often, we hear that there’s a percentage chance of something happening. For instance, there’s said to be a 50% chance of flipping a coin and getting heads. However, in math, probability is measured with values between 0 and 1. If there’s no chance the event will happen, the probability is 0. On the other hand, if the probability is 1, we know it will happen for sure. So, mathematically, the probability of getting heads in a coin toss would be ½. For example, if we calculate the probability of rolling an even number on a die, the result is 3/6 = 1/2, as there are three possible even numbers (2, 4, and 6) out of a total of six possibilities. This is known as: probability equals the number of favorable outcomes divided by the total possible outcomes. So, could we analyze the probability of having a “good hand” in mus? Let’s take a look.

THE RULES OF MUS

In mus there are four players and each receives four cards. According to the Fournier rulebook (if playing with 8 kings), the objective of the “game” is to equal or exceed 31 points by adding up the value of each card. The best game is 31 points, followed by 32, then 40, descending to 39, 38, 37… all the way to 33, which is the worst game. If no one reaches 31 points, then the game is decided by the “point,” meaning the winner is whoever gets closest to 30 points.

As for card values, each card from ace to seven scores its face value, except for the twos, which are worth one point (the same as an ace), and threes, which are worth ten points, just like the king, jack, and queen.

The points are tracked with the help of amarracos, and mus is typically played in pairs. Betting money or not depends on the context and each player’s confidence in their skills!

PROBABILITY IN MUS

A “good hand” is, for example, receiving three cards each worth 10 points, as the fourth card then guarantees that you’ll have a playable hand. Moreover, if the fourth card is worth 1, you’ll have 31 points, the best game. So, what’s the probability of getting three cards worth 10 plus one card worth 1? To find out, we calculate the number of favorable outcomes—how many ways there are to get those four cards—and the total number of possible ways to deal four cards from a 40-card deck.

For the favorable outcomes, let’s calculate how many ways we can get three cards with a value of 10 points plus one card that is either an ace or a two. We’ll focus on kings and king equivalents (threes): out of a total of eight kings and threes, we want to get three. There’s a mathematical tool for calculating combinations, where 8 elements are taken 3 at a time, expressed as the binomial coefficient (8 3) = 56. We also want the fourth card to be an ace or a two, which gives us eight options, so we multiply those 56 combinations by 8, giving us a total of 448 ways to get the desired four cards. That sounds like a lot, doesn’t it? But we shouldn’t be fooled, as we don’t yet know if that’s a lot or a little without knowing the total number of ways to deal four cards.

To calculate the total number of ways we can receive four cards from a deck of 40, we again use the combination formula:(40 4)= 91.390.

Now we calculate the probability of getting these specific cards with the formula: number of favorable outcomes divided by total possible outcomes: 448/91.390=0,0049. So the probability is 0.0049, or about a 0.5% chance of getting three tens and a one… not so high, is it? But don’t worry, there are other ways to reach 31 points, and you don’t need 31 points to win a mus game!

In fact, now that we know how to calculate probabilities, we could calculate the probability of getting any other “good hand.” For example, getting 40 points. Want to try calculating it?

THE ENDLESS POSSIBILITIES OF MATHEMATICS

We’ve seen that mus combines strategy and luck. But understanding the probabilities of certain card combinations allows us to make more informed decisions, which can tip the balance in our favor.

Math enables us to go further in probability calculations, accounting for other possibilities. We know that winning at mus doesn’t depend solely on our hand but also on what the other players have. We might love to predict our opponents’ cards, but that remains uncertain. What we can calculate is the probability of winning a game based on the cards we hold. These probabilities are known as conditional probabilities and follow the most important theorem in probability theory: Bayes’ theorem. Though introduced over 200 years ago, this theorem is the foundation of advanced artificial intelligence techniques, such as the algorithms behind the famous ChatGPT that recently appeared in our lives.

So, the next time you sit down to play mus, remember that behind every move and decision, there’s a world of probabilities you can use to your advantage. Good luck!