Probability & Statistics and the game of dice -MDS 2024#6

These days I have a really high motivation in revising probability and statistics, and it’s surely not a coincidence! I had a chance to play a board game named Minivilles (French version of an orginal Japanese game whose name is 街 コ ロ - Machi Koro, translated as “The City of Dice”). This is a strategic game whose progress is strongly dependent on chance (in daily life terms, luck). The gameplay is simple, yet interesting and probably addictive, especially for those who are into this kind of playing. To sum up, a player rolls a dice to earn coins and builds construction cards, whose number is marked from 1 to 12. The goal is to earn coins as fast and as much as he can to be the first one who finishes building the final 4 landmark cards. It sounds like monopoly game at this point, isn’t it? In the following paragraphs I’ll explain in details for you to see how mathematically interesting the game is and the application of mathematics in life, as well as how real life problems inspire us to study and develop mathematical solution.

Figure 1: A glance at various cards in the game. The yellow cards with a towering symbol and without a number along their name are called landmarks, while the rest are construction cards, numbered from 1 to 12, and are used to earn coins.

The game has some other rules and twists from the special effects of cards, yet in this writing, only the math aspect is mainly discussed, which inspired me to try the game and share with you my experience. As mentioned above, each card is marked with a number from 1 to 12, corresponding to the results of rolling 1 or 2 six-sided dice. There are also cards with 2 consecutive numbers, see Figure 1. My first impression was that these double-number cards are the most powerful cards because they have a higher chance to hit, logically. After several matches, I noticed an interesting point of math-related links in this game. Starting with a dice of 6 sides, the probability (chance) to get a certain number, from 1 to 6, is 1/6 (~17%), and the double-number cards double this probability to ~34% because the occurrence is two times higher (2/6). The game would be too easy at this point. Moving to 2 dice, the game brings the strategic playing to the next level. The probability of getting a certain number from 2 to 12 would be identically calculated as the case of rolling 1 dice, only if the dice has 12 sides, which is not a common case and definitely not the kind of dice that is used in this game. Since there are 2 six-side dices that give a total of 36 possibilities of additionally forming 11 different numbers (number 1 can’t be reached with 2 dice), the chance to get a certain number is not the same as others. Let’s take a look at the Figure 2 to see all 36 possible combinations in a better view (thankfully the possibility set is still relatively small for visual listing, otherwise we would need a formula or even an advanced math model to cope up with the size).

Figure 2: 36 possible combinations of rolling 2 six-side dices.

The probability of getting a certain number, from 2 to 12, is not equal between numbers. Number 2 (each dice gives 1) or 12 (each dice gives 6) have 1 possibility over 36 so the probability is 1/36 (~3%), the lowest in the set. Number 7, in the other hand, has 6 possibilities ([1+6], [2+5], [3+4], [4+3], [5+2], [6+1]) that lead to the probability of 6/36 (~17%), the highest in the set. Following this basic calculation, cards with 2 numbers still always give higher chances, yet not anymore 2 times of increment. The probability of these 2-number cards is simply the addition of probability of each number. For example, card [9-10] has 4 possibilities for number 9 and 3 possibilities for number 10, so its total probability is 7/36 (~19%). Similarly, the card [2-3] has the probability of 3/36 (~8%). Another interesting observation is that the probability starts to decrease in numbers that go away from 7 (like the symmetric property of a matrix), with a constant step of 1/36 (~3%). Refer Figure 3 for the probability of all numbers.

Figure 3: The probability of all outcome numbers with 2-dice rolling.

Are you still with me? That’s all for the calculation. The best strategy I learned from the game Minivilles is that I should aim the number that gives the most chance to earn coins, either card with number 7 or double-number. Yet, the game is quite balanced due to coin distribution and special effects. The card with the highest chance is not the card that earns the most coin, and with combo stack effect on lower chance cards, almost every card has its own power to help you earn sufficiently coins and win the game. Also, the game is well done in the way that dice choice would change the strategy and the progress of the game, which also can make a twist and inverse the leading/the lowest player situation. All of this would challenge each player’s strategy and force them to adapt his/her planning to the progress of the game among other players.

As you can see, understanding the game rules helps you win easier. Do you notice what number 7 has the highest chance? It’s reasonable to call it a lucky number thanks to its probability property. I also use this property in Monopoly. Whenever I see someone about to land on my property in 7 steps, I would build more houses and hotels on that property to make use of the chance of collecting more money in a go. The more money I can collect, the more likely I win sooner or earlier. Before applying this strategy, I played this game with an innocent mind and a pure joy. I still keep that in mind, yet with a good statistical strategy which plays an important part in this kind of game, I can also win more often not just because of luck, or in other words, I choose higher winning probability in every move I make. Each game has its own rules and best way to play, so by studying well how it works and using a suitable set of moves, we can be the master of the game and make dice game less likely a purely lucky game.

Back in the time when I was in high school, I didn’t have much inspiration to learn a subject, particularly math. It was much more like an obligation that I need to try and pass the examinations, year after year. I wish that I could have my current mindset to study better at school. It’s still ok though, I did quite well at school and I didn’t miss too much. Above all, learning is the progress of a lifetime. I would continue to learn and make myself a better version every day. Hope this writing inspires you to pay more attention to maths in particular and to whatever you learn in general. “Invest in yourself. It’s the best investment you can make,” one says. Whatever you learn will be useful in one way or another.

“Stay hungry, stay foolish!” – Steve Jobs.

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About My Daily Stories (MDS):

A mini series of posts telling the life stories that I experience, often related to the moral aspect and I want to dedicate a place to honor nice gestures and kindness in life. A pretty flower can make a bright new day for one or more people in this crazy world. No one can teach anyone else about morality, but we always have the choice to be as kind as possible. I will also narrate in 3 languages to have multicultural perspectives because this is the environment I live in (a Vietnamese living in France loving to speak English 😁).