A Room, Two Dice, and Some Ammo...

                                                                   

People seem to especially like puzzles and paradoxes so I’ll re-run another that I haven’t re-visited for awhile. I like this one because it is less well-known (also a bit more grisly!) than most of the best paradoxes out there (like “Two-Envelope,” “Sleeping Beauty,” or “Newcomb’s paradox”).  It is the “Shooting Room paradox.” Here’s how I set it up last time I posted it:

---------------------------------

The situation involves a theoretically infinitely large room and infinitely large population of players… and, 2 dice. The first 'player' enters the room and the 2 dice are thrown. IF the result is double sixes, the player is shot and game over. Otherwise the player leaves the room unscathed and 9 new players enter. Once again the dice are rolled, and IF the result is double sixes, ALL 9 are shot. If not, they leave, happy and healthy, and 90 new players enter the room….

This pattern continues, with the number of players increasing tenfold with every new round of play. The game simply goes on UNTIL double sixes ARE rolled and an entire room group is shot, at which point the game is over.

IF you are in the pool of players how worried should you be for your safety? ...perhaps not very, since your chance of being shot is NEVER more than 1 in 36, or < 3% (the chance of double sixes).

BUT, your wife discovers you are in a group about to enter the room, and she is petrified, because she understands that inevitably ~90% of ALL players who participate will be shot before the game is over! Who is perceiving the odds correctly? (i.e., the overall likelihood of death for participants in this "game" is 90%, yet the chance for any single individual is 1/36).

---------------------------------

A solution can actually get into some heavy mathematics (but also some semantics) that I’ll bypass opting instead for a simple, basic explanation that the 90% figure is of course an aggregate figure, and even though it's eventually true and accurate, once the game is over, it doesn't change the real in-the-moment probability of harm for any single individual, which is only 1 in 36 ...in a sense, one cannot invoke reverse causation in adjudging the probability at a given prior point-in-time. There are, by the way, other versions of the paradox set up with less grisly, more game-like, scenarios.