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.