# 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:

# 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.

• The probability of player K-1 (prior player) reaching step Y (with Y < X )
• The probability of player K taking X-Y steps

# 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:

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Principal Data Scientist, Decision Science @ Atlassian

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## Alejandro Martinez Vargas

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Principal Data Scientist, Decision Science @ Atlassian