What we have here are two bits of card.
On the bottom of it I'm going to ask you
to write down two different numbers,
they can be any numbers at all they've
just got to be different. And I've
actually called them for you already A
and B, but they're two different numbers.
(Brady: So it could be- could be a million, it)
(could be 10?) - Yeah it could be 0.001,
it could be 1/2, it could be pi or e or
anything like that - just two numbers. And
you're going to put them face down and-
oh you can't have imaginary numbers,
they've got to be say real numbers. What
I'm going to do, I'm going to just decide
to overturn one of them, so this one here
say, and I'm going to tell you with more
than 50% chance of me being correct
whether this one is bigger, the upturned
one is bigger than the one that's
facedown. Now you would have thought that
this would be absolutely impossible
because surely, you know, I only have a
50/50 chance. I've got these two numbers
here, one's bigger than the other and
I've got a 50/50 chance of picking each
one, you know- we obviously put A and B
but they can have two different values.
So depends what you've put as A and B, it
depends which one I- chance is just
50/50. But there is a fantastically
ingenious way that puts the odds in my
favour; that means I will win this game
over half the time. What I'm going to do,
and the way that this works, is that I
myself I'm going to generate another
number completely randomly and I've done
one already and I'm going to call it K.
Okay? So what I'm going to do now is I'm
going to look at the number that's been
revealed - A. If the number that's been
revealed is bigger than my random number
I'm going to say A is the biggest number.
If A is smaller than K, the random number,
I'm going to say no no this one here, the
one that's unrevealed, the hidden number,
is the biggest number. And just by doing
this I can guarantee that statistically
my odds are much- well, are better than 50/50 so
it's more likely than not that I will be correct in determining
whether the revealed or the hidden card,
which one is bigger. K is a randomly
generated number, you know the mathematical
small print so to speak is that I need
to take it from a distribution that
covers all the numbers. So for example
the normal distribution, you know you can
set the the mean, the standard deviation,
it just needs to come - and you get this on the internet -
I just need to generate a number which
could possibly be, there is a chance no
matter how small, of it being any number.
And we can do this fine- so know this
is K. - (Brady: if we were playing this game)
(for real Alex, how would you have generated K?)
(Would you have gone over to your)
(computer or?) - Yeah I would have gone over to
computer, I would have googled random
number generator normal distribution; I
would have put the mean, the standard
deviation, and I would have got something.
And because, you know, we're humans with-
you could be clever and cleverer and
say this is going to be a googol, this is
going to be a googol times a googol;
most people are going to say, this is a number
between one and a hundred, this is
probably a number between one and 100,
and so I would put with my- you know
the mean would be 50 and you know
standard deviation 20 - something like
that. So why does this work? Now let's
consider values of K. And in fact what
I'm going to do, I'm going to reveal this
A and B. There are three different
possible scenarios: either K is less than
both A and B; or K is bigger than both
A and B; or I have managed to get K- I've
hit the spot, is between A and B - or it
could be like that 'cos we don't know
which one's bigger. Basically K is in- is
in between. And so what I'm going to show
now - put this down - is that if K is lower, as
I said I'm going to choose A, always I
stick with A. And because we know that
both of these numbers are bigger than K,
because K is lower, it's still a
50/50 chance. if K is bigger I flip. So in
both cases where my number K that's
randomly generated is either less than
both of A and B,
or bigger than both of A and B, then the
odds are still 50/50. When it makes a
difference is just say that K is in
between A and B; because if K is in
between A and B I always win.
I always know which is biggest. So if I
say A is biggest then I know I'm right
because K has to be less than A, we know
that K is in between, so I'm right. But if
A is less than K, we know that K is in
between A and B so I will be choosing B and
I would also be right; because B is
bigger than A. So just by this random
generation of a number, that has
absolutely nothing to do with the
numbers that you chose and you wrote here, if it
so happens that this number K is in
between these two numbers both scenarios
I am correct. - (Brady: Alex it feels like the)
(bigger the gap between A and B that I)
(left when I wrote down the numbers to)
(start with, the greater I was increasing)
(your chances?) - Correct. So if you knew that
I was doing this you could try and make
it as difficult as possible for me to
get the edge by choosing an A and a B that
are really really close together. But so
long as I'm choosing from a distribution
where all numbers are game, there's a
slight chance of choosing any number, I
have the edge, I always have the edge.
Obviously if you choose big numbers- you
know, if you were to choose minus 1000
and plus 1000 I would pretty much
win every time. So yeah the larger the
gap the easier- you know, it's like the
wider the goal, to use a football
metaphor, the wider the goal.
...I said I
haven't actually done this as an
experiment, but imagine if I was to tell
you put two numbers down; you're probably
going to say something like- I don't know 12
and 30? You know, that's kind of quite a
reasonable thing that you're going to do.
Most people would probably choose not a
digit, that seems a bit too small but in
the hundreds that's just like too
complicated. You'd probably do something
between say 10 and 50. So if I'm- you know if I'm
put my distribution so that it's heavily
picking up those numbers between 10 and 50;
I'd say that you play this game a
hundred times, maybe there's someone who
actually wants to run the simulations, I
wager that it would actually be- you'd be
winning not just more than fifty but
a significant amount more than fifty
percent of the time. - (Brady: I guarantee you one)
(thing Alex, I guarantee you there will be)
(Numberphile viewers who run this)
(simulation.) - I can't wait! I can't wait!
Even if I proved wrong!