So I want to tell you about a classic
paradox in probability that is sort of a
cautionary tale for students when
they're about to hear a puzzle or
they're setting up some kind of problem;
Bertrand's paradox. There was a person
who wrote this probability textbook, last
name Bertrand. He mentioned in there why
you have to be very careful with
probabilities when infinity is involved.
So what I'm going to do is have a circle
and
I'm going to draw a random chord
somewhere in that circle. So a chord is
something that just connects two
different points on that circle right?
What I want to ask is a certain question
about the distribution of the length of
this chord. You know, there's some kinds
where if it just glances through the
circle that length will be very short, if
it went through the diameter that length
would be very long.
One of the most natural questions you
might ask is something about the mean of
that length; but that turns out to be a
little on the messier side for the
cleanliness of what Bertrand wanted to
do. So instead he says, compare that
length to the length of
the sides of an equilateral triangle
that's inscribed in that circle, which
might seem kind of weird, like why are we
talking about triangles? But it turns out
to make the answer especially clean. So
if we take some equilateral triangle
where - forgive the fact that I'm a little
curved in the side length there - and we
call this side length something like s.
What Bertrand asks is, what is the
probability that the length of your
random chord is bigger than the side
lengths of that triangle? And we could
get a little numerical about this: if
we say that the circle has a radius of
1 then geometry students might
remember their 30, 60, 90 triangles. This
angle here is 30 degrees,
this angle here is 60 degrees, and so
this side length will be exactly one
half,
this side length will be square root of
three halves. So we could also say if
it's a unit circle what we're asking
about is the probability that the length
of our chord is bigger than the square
root of 3? So just a simple question
like anything else you might come across
in some books, some list of exercises at
the end; and let me show you how to solve
it, it's really not that hard. Without
loss of generality you can assume that
the first point is one of the corners of
the triangle, if it wasn't we could kind
of rotate the triangle so that was the
case.
And then we weren't given any specifying
information about the nature of the
chord, so we'll assume that the next side
is uniform somewhere around the circle.
If the second point chosen was over
there we would have a chord that's
shorter than the side lengths of the
triangle. If it fell into
this kind of critical region on the
opposite side of the triangle,
up here,
that region
then the chord that you draw would be
longer;
and then similarly if it was on this
other side it would be shorter. So
probability student, they're doing their
homework, they say what's the probability
that this is longer? Not a problem,
it is going to be exactly 1 in 3
because the circle's evenly divided into
three pieces, there's only one of
those pieces that will give you longer
chords, said and done not an issue. - (Brady: So if)
(any random chord, there's a one-third)
(chance it's going to have that long-) - Yeah,
yeah. No paradox to be found, in fact
let's verify this empirically. I ran a
simulation. Essentially what I do is I
choose a random pair of points somewhere
in the circle, I connect them and then
I'm going to count
whether or not that ends up longer than
what we need, and then basically ask the
proportion where that's the case. And
I'll color it blue whenever it's longer,
and then white whenever it's shorter. I'm
just going to have to simulate a whole
bunch, it'll do on the order of a
thousand. And this ratio will be telling
us what proportion of them are longer
than the length we care about, and yeah
it seems like it's going roughly on the
order of one third so 0.33, kind of
oscillating 0.34. So we did it
theoretically, we did it empirically, we
have an answer. So another student comes
in, they're doing the same homework and
they say, you know the funny thing about
chords is that they are uniquely
determined by their midpoint in the
circle.
So they say,
if you give me a midpoint of the chord
they say actually there's one and only
one chord
for which that would be the midpoint. You
draw the line that's perpendicular to
the radius between that midpoint and the
center of the circle
and that will give you one and only one
chord. So then they go to their triangle
and they try to reason, when is it the
case that our chord is going to be
shorter than the side lengths of that
triangle?
And
what they can
also do is come up with a pretty
geometric argument where they think
about the circle that's inscribed
in that triangle - got sort of a deathly
hallows vibe going on here. And what
they're able to conclude is basically
when the center of the chord is on the
inside of that inscribed circle
then
the relevant chord is going to end up
longer than the side length of the
triangle, we can kind of see that there.
But if the random midpoint that you
chose was on the outside of that
inscribed circle the chord you find will
be shorter.
And
we simply say, okay, what is the size of
that circle?
And
we actually drew one of our lengths
earlier and concluded that it was one
half. So we know this inner circle has a
radius which is one half. So that inner
circle has an area which is
one-fourth, and you know, the
probability of getting the chord that's
longer is the probability of your
randomly chosen midpoint falling into
that smaller circle. So the answer to our
question, that Bertrand posed,
that you know l is greater than s
is actually equal to one fourth. - (You're right!)
Student circle's their answer, they turn
in their homework and they say, for extra
credit I'm going to simulate it. I'm
going to choose a random point somewhere
inside this circle and each time that I
draw one of those points I'm going to
draw the unique chord that has that
point as a midpoint. I'm going to count
up how many cases did I get where the
chord was longer than the triangle point?
How many was it shorter? See if it's
roughly one quarter. And they do it and
as they get around a thousand samples
they see that, yeah, this is you know it's
around 0.25. By randomness it looks like
it happens to be a little bit above
right now but certainly within the
threshold you might want for a
statistical distribution. They also get
this kind of spider web pattern on the
inside; although it does look a little
bit different than our first students
where
you might notice the spiderweb is a
little bit sparser in the middle of this
one that we have in comparison to the
older one that we had, the first student
simulated it and things seemed a little
bit more uniform in there. So it makes
sense they got different answers, there
was a different distribution at play. And
so that's interesting that these two
seemingly reasonable methods give you a
different answer.
But then this class actually has three
students, and a third student has a
different way of thinking about it, and
maybe they'll be able to break the tie.
And the way that person thinks about it
is-
you know the the chord that we draw,
it's going to have kind of a random
angle. The angle that it has between 0
and pi or 0 and 2pi, however you want to
think about it, there's no reason to
prefer any one of those. So the radial
line that it's intersecting
should just be chosen uniformly around
the circle.
And then there's no reason to think that
the midpoint of your chord should be, you
know, biased towards being on the center
of this radial line or biased towards
being the end, it should be chosen
uniformly along this radial line.
And so what they want to ask is, if we're
choosing something uniform along this
radial line, what's the probability that
the corresponding chord is longer than
the length we care about?
And again you can kind of do a little
pretty geometric argument here
where,
if we've got our triangle,
same conclusion we had before where the
length from the center of the circle
directly down to one of the legs of the
triangle perpendicular
is one half.
And so-
I drew it a little bit poorly here where
this side looks like it's shorter but
that's that's a function of my bad
drawing, if you do it more exactly. This
portion where your chords are going to
be longer than the relevant triangle
side length, you know, if I choose one
here draw the chord,
you can just see it ends up longer than
that triangle length - that should be
one half of the time.
And again, you know, they want to say hey
I'm gonna check my homework, I've learned
that sometimes I can't trust my
theoretical results, and the way they do
it is they choose a random radial line
which I've colored in yellow then a
random point along that line that I've
given this sort of glowy yellow dot. And
they do that a whole bunch of times,
choosing a random line, choosing a random
point along that chord;
their expectation from their geometry
and their theoretical reasoning here
would be that this comes out to be
around a half, but they want to run a
thousand simulations just to see if that
happens to be roughly true. And look! So
what they got actually is really
close to one half,
and it feels pretty uniform. You know, you
look at the spider web it seems nice and
uniform.
So this person they go they circle their
answer and they say,
Mr Bertrand- Professor Bertrand I've done
my homework and I know with confidence
now that the answer is one half.
(I was convinced by all of them.) - They all
seem pretty reasonable. So Bertrand's
conclusion here was you have to be
careful when probabilities with an
infinite space are involved,
because you don't know
simply when someone says choose a random
blank is that a well-defined notion?
Because here I can have something where
I say, choose a random chord; seems like a
reasonable question to ask,
and we've got three reasonable
approaches that give you these different
answers. Now this is a little bit
worrying, because all the time in
probability you start off some question
or some model of the world by saying
something like, you know, I want to choose
a random number between zero and one.
I say choose a random number
and it's implicit that it's going to be
some kind of uniform distribution and
it's unambiguous what you mean by that
randomness. Or you say, hey I've got a
circle, I want you to choose a random
point on that circle.
Or I've got a sphere, you know, and I want
you to choose a random point somewhere
on the surface of that sphere. You know, I
do this all the time in videos that I
make, I start a puzzle I say choose four
random points on the surface of a sphere,
what's the probability that yada yada;
the tetrahedron that connects them
contains the center or some interesting
puzzle you want to get to. Does the
premise even make sense, right? Is it the
case that when you say choose four
random points there aren't two seemingly
reasonable ways to go about that that
turn out to be
distinct in that they give you different
numerical answers, but seemingly
reasonable to each person who did it. The
video I'm making right now, I say choose
a random orientation of an object in 3D
space and then yada yada question. How do
we know it's reasonable to say choose a
random orientation? So this is a little
bit worrying that Bertrand could come up
with this example where we get this
ambiguity.
So there's Bertrand's paradox, but you
can probably guess Grant's got plenty
more to say about it. Check out our
second video, even more about Bertrand's
paradox, where we go deeper. - But the
unambiguous thing is honestly what I
would have thought would be the- the
worst of the three methods. - This was also
made as a companion to Grant's own video
about the average area of a cube's
shadow. You'll find links to that in the
usual places and of course on the
3blue1brown channel.