Vsauce, Kevin here -- with a deadly dice game
that from your perspective gives you a 97.2% chance of surviving. But from your friend Billy’s perspective,
there’s a 90% chance that this game kills you. How does this make any sense? Well, here are the rules of our theoretical,
potentially horrific game. One person walks into a room and rolls a pair
of dice. As long as they don’t roll double sixes,
they walk out of the room alive and they win one million dollars. The game continues but this time 10 people
walk into the room. And an appointed roller rolls the dice and
if they don’t roll double sixes, all 10 people walk out of the room alive and with
a million bucks each. Each round the number of players in the room
increases by a factor of 10. So 1, 10, 100, 1,000, 10,000 and so on. Each participant only gets to play one round,
and as soon as double sixes are rolled, the game is over forever. Also for this game we have an unlimited supply
of money and infinity people because we’re gonna get to 10 billion people by the 11th
round and that’s more humans than we have on Earth. Oh, we also have a room that can hold infinity
people. Suspend reality. When an unlucky roller finally rolls double
sixes -- also called “boxcars,” or “midnight” -- everyone in that room dies instantly. Kind of a grim game but a million bucks is
life-altering money so you may determine that it’s worth the risk. After all, the odds of survival are in your
favor. Six sides on each of two dice, marked with
dots officially called pips, means there are a possible 36 combinations you can roll. That's a 1 in 36 chance you’ll roll double
sixes, which is only 2.77% repeating which we’ll round up to 2.8%. To put it another way, there are 35 different
combinations of survive rolls and only 1 combination of death roll. You might think that the game could be played
for 18 rounds before there was a 50/50 chance of rolling double sixes because 18 rolls out
of 36 possible combinations is 50%, right? Well, no that's actually not how we figure
this out. In fact, you’d have to roll 25 times for
a 50% chance of rolling double sixes thanks to combinatorial probability. The easiest way to figure this out is actually
to put the problem in terms of success instead of failure. So there’s a 35 out of 36 chance that our
roll will let us survive. To calculate the probability of the game-players
all having a certain number of surviving rolls, we raise (35/36) to the nth power, where n
is the number of rounds the game is played. In 5 rounds, the chances that everyone survives
are about 87%. 10 rounds and that drops to 75%. At n = 18 rounds, there’s still a 60% chance
that no one has rolled double sixes. It isn’t until the 25th round that the odds
shift against us. I mean, you COULD roll double sixes on the
very first try... it's just not likely. But each player is only playing one round,
and we know that 1/36 is about 2.8%. Which means, to you, the player, you have
a 97.2% of living and getting the money whether you’re on your own in the first round or…
in Round 9 with 99,999,999 other people. But when your friend Billy hears that the
game has finished and that you were one of the players, he knows there’s about a 90%
chance you’re no longer in the mortal realm. And he’s understandably concerned. After all, he is your best friend. The reason he's concerned is very simple. Despite just a 2.8% risk of a dice-rolling
fatality, there’s about a 90% chance that you were one of the unfortunate, recently-perished,
losers. The danger is a matter of perspective. You know you’ve got over a 97% chance of
surviving your round and getting paid. Your risk of death by double sixes is low. But poor, sad Billy realizes that because
of each round's escalation of players, the final group that rolled double sixes and succumbed
to the deadly rules of the game, makes up almost 90% of all of those who played… so
if you participated, you were likely in that group. To Billy, there’s only a 10% chance you
survived. Your truth is that your chance of surviving
is extremely high. Your friend’s truth is that the chance of
you being dead is nearly 9 out of 10. In the double sixes death game, reality doesn’t
change -- but a person’s perspective can flip that probability on its pips. And as always, thanks for watching. Um, I'm actually gonna go through all the
combinations right now um just on the.. on camera. Okay so. So we have one one. One two. One three. One four. One five. One six. Two one.
is it perspective or is it that "chance of losing this particular round of an ongoing game" and "chance of having been in the last round of a completed game" are just different quantities? The probability of survival given being a specific round is high; the probability of survival given having been in the game is low. Most rounds are won, but most people are in the losing round.