🦑 Squid Game — What are the chances to survive the Glass Bridge?
Have you watched Squid Game? Probably yes, as it’s become the most popular show ever on Netflix… But if you haven’t, stop, 🛑 SPOILERS AHEAD!! 🛑
On 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 2 glasses the player can step on. One of the glasses is safe but the other will break and the player would be ‘eliminated’.
I don’t know about you, but since the introduction of the game I was thinking what were the odds to survive and how many where expected to cross the bridge safely… 🤔
How many can we expect to survive?
My first thought was how many steps each player is expected to ‘unlock’ for the rest of the players. Even if a player steps on the wrong glass, it will unlock one step for the next player in line. So that’s 100% probability to unlock at least 1 step. Of course, there is 50% chance to select the safe glass, in which case, the player would be also 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 players), 12.5% chance to have 3 safe jumps etc… So basically we can calculate the expected steps each player will unlock, and it looks like:
Nice, we all recognize that summation from our school days! 🎉 And we all remember it’s equal to 1, don’t we? 🤨 With that, we know that 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… 🥴
Ok, but what are the probabilities to survive for each of the players?
It’s clear that is not the same being the first player or the last one… Let’s try to compute what are the survival chance for each player.
Player 1 would need to pick the right glass 18 times… Each of the jumps have a 50% chance, so the probability of Player 1 reaching the other side is
That’s 1 in over 260.000! Sadly not very good odds for Player 1…
Probability for Player 1 is relatively easy to calculate, but things start to complicate for the others… Their chance of survival depend on how well or how bad prior players 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
I will not go through all the calculations here, but if you are interested you can find them in this doc. With that data, we can see the initial probability of each player to reach the end of the bridge alive!!
When we mentioned 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!
Hmmm, not very fair game… how can we maximize the probabilities for each player?
During the presentation of the game, the rules said that they have to go in order. However, during the game, one of the players refused to go and forced the next player to take the next step… If that’s allowed, the players can think of different strategies to maximize the individual chances to survive. One thought is that Player 1 jumps step 1, player 2 jumps step 2 and so on… That way they ensure each player does at least 1 jump. But there are 2 more steps… For those 2, I would keep the order, so if player 1 survived, then player 1 take step 17 etc… With this approach, these are the new survival probabilities:
It improves significantly the chances for 9 of the players! And actually it improves the average survival probability from 43.8% to 45.3%. Although I’m pretty sure P10-P16 wouldn’t be very happy with this approach… and you try to convince #101 about taking this strategy!!