South Korean survival drama television series ‘Squid Game‘ has taken the world by storm ever since it was released in September. The series is centred around 456 debt-ridden players from different walks of life. They are invited to a tournament to play a set of children’s games with deadly outcomes for losing a round in their quest to win a ₩45.6 billion prize.
In episode 7, the 16 players that are still alive play the Glass Bridge game. The rules are simple: 18 steps to the end of the bridge, each step with two glasses the player can step on. One of the glasses is safe, but the other will break, and the player would be ‘eliminated’.
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Alejandro says each player is expected to ‘unlock’ for the rest of the players. Even if a player steps on the wrong glass, it will unlock 1 step for the next player in line. So that’s a 100% probability of unlocking at least 1 step. Of course, there is a 50% chance to select the safe glass, in which case, the player would also be unlocking the next step (as they will do a second jump). Then, 25% chance (50% x 50%) to have 2 safe jumps (unlocking 3 steps for the rest of the players), a 12.5% chance to have 3 safe jumps, etc. So, we can calculate the expected steps each player will unlock, and it looks like:
Each player is expected to unlock exactly 2 steps. There are 18 steps, so we expect 9 players to unlock them, and therefore we expect 7 players to survive. In the show, only 3 players survived, so they didn’t do very well.
Player 1 would need to pick the right glass 18 times. Each of the jumps has a 50% chance, so the probability of Player 1 reaching the other side is:
That’s 1 in over 260.000! Sadly, not great odds for Player 1.
For the others, their chance of survival depends on how well or how bad the players before them do. The probability of player K reaching step X depends on two things:
- The probability of player K-1 (prior player) reaching step Y (with Y < X )
- The probability of player K taking X-Y steps
As we know that Player 1 had it difficult, we might have thought that Player 5 for sure was going to have much better chances, but not really! Player 10 is the first player with more chances to survive than to be ‘eliminated’.
And this chart also proves what we mentioned earlier: we expect 7 players to survive!
In the past few weeks, the show has earned rave reviews and has become a pop-culture phenomenon with memes, facts, references, and theories on every major social media platform.