Ok, so you remember the setup: 100 people
have had marked dollar bills put randomly into 100 boxes, and every person, one at a
time, gets to look in fifty of the boxes, and you all collectively win only if every
single person finds their own bill. $100 dollar payout to each of you if you win,
nothing if you lose. In the last video I asked whether or not you
should take the bet. Did you make the right choice? I mean, clearly, if all 100 of you gamblers
just choose 50 boxes at random when you go into the room, the likelihood of each of you
finding your bill is one half, so the probability of ALL of you finding your bills and winning
the whole game is one half times one half times one half… a hundred times. And one half to the power of 100 is about
0.0000000000000000000000000000008. Basically zero. You're more likely to roll 9 one-roll yahtzees
in a row - and then pick the king of hearts at random out of a deck of cards. You could do a little better by trying to
correlate which boxes you pick - like, if there were only two people and two boxes,
they should each pick different boxes, because if they pick the same box only one of them
can possibly find their own bill, and they'll automatically lose. But this kind of strategy only improves your
odds a tiny little bit, and it has less benefit the more people there are. However, despite all this, it turns out that
there's a strategy that results in you gamblers winning over 30% of the time! This strategy is SO GOOD that you're more
likely to win using it than if just TWO people out of a hundred picked randomly, since one
half times one half is twenty-five percent. What we’ve forgotten is that you can obtain
information FROM the boxes themselves, because the room is exactly the same each time. The first box will always have the same-numbered
dollar in it, and so will the second box, the third, the fifty-seventh, etc. All you have to do is use the dollar in one
box to tell you which box to go to next. This might not seem like a great advantage,
but it's HUGE. The trick is to start with the box that corresponds
to your number. Your dollar probably isn't IN that box, but
the bills inside will take you on a random path through the boxes, a scavenger hunt where
each box tells you where to go next. You can think of this as linking up boxes
in chains, connecting them based on the bills inside. Any random arrangement of the boxes results
in a different set of chains, some long, some small. Short chains are ones like "box 30 says go
to box 82, box 82 says go to box 5, 5 says go to 30." That's a chain of link 3. Long chains link up more boxes before circling
back - and the LONGEST possible chain links up all 100 boxes. But each chain ALWAYS circles back, since
there are finite boxes. And of course, if you end up back where you
started within 50 boxes, you've won, because that means the previous box had a dollar bill
that told you to go your starting point, and you started on your number, so the previous
box's dollar bill was YOURS! And that's why this strategy is so good - because
when the boxes are randomly arranged, about 30% of the time there will be NO chains connecting
more than 50 boxes, so not only will you win, but so will everyone else - as long as they’re
following the same strategy! The magic of this solution isn't that it's
better odds for any single individual to find their dollar (it isn't) - it's that all of
the successes and failures are forced to happen together - everyone wins together, everyone
loses together - your fates are linked by using the numbers inside the boxes to guide
you along invisible chains… towards victory.
he's speaking too fast
Dear god....he's almost as bad a Vi Hart.