Calculating the Probability of Odd-Even Marble Game in Squid Game

The death of the Old Man made episode six (gganbu) the most memorable throughout the Squid Game series. Moreover, this Squid Game episode also revealed all the true evil nature of humans and their survival instinct, even when their opponent is their own friend. The game played was simple - marbles, where pairs of players could freely determine their own game rules.

Each player was initially given a capital of 10 marbles each and given 30 minutes to play against their partner without violence. The player who successfully obtained all ten of their opponent's marbles was declared the winner and could proceed to the next game.

In my previous writings, I have calculated the probability of Nobita getting zero scores to calculating the probability of winning UNO. I will play with probability again to calculate the odd-even marble game.

This game was played by most player pairs in the gganbu episode of Squid Game. Odd-even was played by Gi-hun (main character) with the Old Man (Il Nam Son), Sang-woo (businessman) with Ali (worker), and Deok-su (thug) with player 278 (his subordinate).

The rules of this game are simple. The opposing player (Old Man, for example) only needs to correctly guess whether the number of marbles in their opponent's grip (Gi-hun, for example) is odd or even. The Old Man places a bet in his grip, where Gi-hun will pay the amount of his bet if the Old Man successfully guesses correctly. Conversely, the Old Man's bet will be taken by Gi-hun if the Old Man guesses wrong.

What is the probability of being right and what is the probability of being wrong in guessing at the beginning of the game?

Before calculating the difficult ones, we should calculate the easy ones first, while refreshing the concepts in probability.

First, the universe. At the beginning of this game, the universe of events $S = \{1,2,3,4,5,6,7,8,9,10\}$ because each player is given 10 marbles as capital.

Second, events. The odd event is $\text{Odd} = \{1,3,5,7,9\}$ while the even event is $\text{Even} = \{2,4,6,8,10\}$.

So, if we summarize at the beginning of the game, $n(S) = 10$, $n(\text{Odd}) = 5$, and $n(\text{Even}) = 5$. Thus, the probability that the marbles in the grip are odd is:

$P(\text{Odd}) = \frac{n(\text{Odd})}{n(S)} = \frac{5}{10} = 0.5$

And vice versa.

In the sixth episode of Squid Game, Sang-woo doesn't trust Ali and says: "How can you keep winning? The probability of this game is 50:50." Will the probability remain 50:50 throughout the game? Of course not, the game runs dynamically and the probability will be influenced by the number of opponent's marbles.

What is the probability of correct guessing if the number of marbles has decreased/increased?

Suppose the first game in Squid Game between Sang-woo and Ali has been played with a bet of one marble and won by Ali. In the second game, Ali's marbles total 11 and Sang-woo's total 9. Here's an illustration of Sang-woo's marbles.

$\text{Odd} = \{1,3,5,7,9\}$

$\text{Even} = \{2,4,6,8\}$

$S = \{1,2,3,4,5,6,7,8,9\}$

Suppose the second guesser is Ali, then the probability that Sang-woo's marbles are odd is:

$P(\text{Odd}) = \frac{n(\text{Odd})}{n(S)} = \frac{5}{9} = 55.56\%$

while the probability that Sang-woo's marbles are even is:

$P(\text{Even}) = \frac{n(\text{Even})}{n(S)} = \frac{4}{9} = 44.44\%$

General Formula for Dynamic Probability

As the game progresses, the probability changes based on the current number of marbles. If a player has $n$ marbles remaining:

• If $n$ is even: There are $\frac{n}{2}$ odd possibilities and $\frac{n}{2}$ even possibilities
• If $n$ is odd: There are $\frac{n+1}{2}$ odd possibilities and $\frac{n-1}{2}$ even possibilities

This creates an interesting strategic dynamic where players with fewer marbles actually become more predictable in some scenarios.

What makes the marble game in Squid Game fascinating isn't just the 50:50 probability at the start, but how this probability shifts throughout the game. The dynamic nature of probability, combined with psychological warfare between players, creates a compelling game that's both mathematically interesting and emotionally devastating.

The lesson here is that even in games of pure chance, probability is rarely static. Understanding how odds change over time can provide strategic advantages - both in fictional death games and real-world decision making.