JAMES CLEWITT: I think this is
going to be a really exciting video, because today we are
going to generate some random numbers with strontium-90-- radioactive strontium-90. I find all of this quite
unnerving because this stuff is dangerous. You have to really respect it. It will hurt you. Let's do it. You've asked me to generate
for you a random number, right? And I want to throw that
straight back at you. I want to play a
game, actually. What I want you to do-- all of
you, everybody, all right? You watching? Everybody, I want you to clear
your head for a moment and I want you to think of a random
number from 1 to 10. Got it? OK. BRADY HARAN: Choose a number
between 1 and 10. MALE SPEAKER: Do
I tell you it? OK. I'll choose 6. MALE SPEAKER: 7. BRADY HARAN: Choose a number
between 1 and 10. MALE SPEAKER: 7 MALE SPEAKER: 9. MALE SPEAKER: 7. 7 is more likely to be chosen. And I'm annoyed. JAMES CLEWITT: I'm learning
that very quickly. MALE SPEAKER: I'm annoyed that
I picked 7, actually. JAMES CLEWITT: 45% of you are
going to have said 7. BRADY HARAN: Choose and
then tell me a number between 1 and 10. JAMES CLEWITT: 7. 7. Everybody says 7. Obviously, you're not going to
choose 1 or 10, because that's not very random. You're not going to choose an
even number, because they don't sound particularly
random, either. And you're not going to go for
5, because 5's in the middle. Hell, that's not random. So you probably want
a prime number. So 9's out. 3 is too small. 3 is like you're not trying. You haven't worked hard
enough to get to it. So what are you left with? You're left with 7. So everybody chooses 7. So is there something innately
random about the number 7? No, of course not. That's ridiculous. There's nothing random
about any number. If I said to you pick a random
number, and you said 5, how do I know that's random? If I ask you do it again and you
came up with 5 again and again and again, clearly, that's
not a random number. The property that makes a number
random isn't in the number itself, but it's in
the sequence of numbers. Any individual number
isn't random. 1-- is the number 1 random? It entirely depends on
the context, right? Just 1 on its own-- there's
nothing random about that. But if I have 1, 5, 2,
8, 7, 4, is 1 random? Yes, 1 is a random part
of that sequence. Or at least, it's as random as
the human mind can make it, which, as we've just seen from
my little game, isn't particularly random at all. As a scientist, I model gases. And as a part of modeling gases,
I need to give them a random kick to get them started
so that I don't model an identical gas every time. I can't go through and give each
molecule a random value from my own mind. I use a computer to choose
my random numbers. And it's spectacularly
bad at doing that. There's no chance of getting
a computer to generate a completely random sequence. It's not a random machine. We have to generate a sequence
of numbers which look and feel random to us. But they're not random. If we look closely enough,
we're always going to find patterns. So the easiest way to go
about doing this-- can I write this down? Let me write this
down for you. BRADY HARAN: Please. JAMES CLEWITT: Typically, the
way that a computer engineer is going to go about generating
a sequence of random numbers for you is
to think of a number. And we'll just call that number
x0, our first number in the sequence, or our zeroth
number in the sequence. And we're going to multiply that
number by something big, something that feels random. And we're going to call that,
rather unimpressively a. Typically, it's going to be
something of the order of tens of thousands. At this point, there's
absolutely nothing random about this whatsoever. Then I'm going to take another
big prime number in the tens of thousands and we're
going to add it on. And with a spectacular lack
of imagination, I'm going to call that b. Still not random. I know it's not random. The next step is probably the
only complicated bit. I'm going to take the number
that we've just got and I'm going do a modulus-- which we
normally define as just being a percent sign-- m. And this is a cyclic division. So essentially, it's the
remainder of a divide. So if I were, for example, to
use m of 8, if I do a cyclic division of 64 divided
by 8, I get 0. If I use 66 divided by 8,
I'm going to get 2. It's the remainder
of that division. BRADY HARAN: So you make this
big number and then you do that process to it using
this number? JAMES CLEWITT: Correct. BRADY HARAN: So you must
still get the same number every time. JAMES CLEWITT: You still get
the same number absolutely every time. And the answer-- we're
going to need an equals sign in here. What we get out of here is the
next number in a sequence-- x1. Then I'm going to take x1 and
I'm going to feed it back in. And we're going to use
that to generate x2. And then I'm going to feed that
back and generate x3. So I get the sequence of numbers
that feel like they might be random. If I were to start with x of
0, then I'm always going to get an identical sequence. So if I'm going to write a piece
of software where I want a random sequence, I need the
value of x0 to be different every time. If I kick my computer off again,
I need it to be in a different state to start with. And the most obvious way of
doing that, and the way I do that, is I simply
use the time. I use the number of merely
seconds since January [? 1970. ?] And then I'm going to
feed that into x0. And then x1's different. x2's different. x3's different. But-- and this is the important
thing-- because this is a cyclic
division, it's going to eventually repeat. And depending on which numbers
you choose for a and b and n, it'll repeat sooner
rather than later. You're going to get
cyclic patterns in your random numbers. Now, we don't call these
random numbers. We call these pseudorandom
numbers because they're kind of random. They're not really random. We can do better than
that, though, right? OK. So if you asked me to generate
a random number-- let's go back to the game we played
at the start. And I'm going to say, let's
generate a random number, this time between 1 and 6. There's a really obvious
way of doing that. I brought a die with me. We throw it and we generate
a random number. And each of those numbers, 1 to
6, has the same probability of coming up. But that's not always true. What if I want to generate a
number between 1 and 12? Let me get another die. First of all, I can't generate
a 1 with two dice. I recognize that. So it's going to be a number
between 2 and 12. We throw that. We get 6. There are many, many more ways
to throw a 7 than there are to throw a 2. There's only one way
I can throw a 2. I've got to throw double 1. If I want to throw a 12, I've
got to throw double 6. But I can throw 7 with a 3 and
4 or a 4 and 3 or a 5 and a 2 or a 6 and 1. There's loads of ways
to do that. So you're far more likely. Let me draw this for you. For one die, this probability
is the same, and it's 1/6. But for two dice, then you're
going to get a probability which is quite sharply peaked. What if I brought
more dice out? I'll just throw three for now. So now, we're throwing-- that's random, right? BRADY HARAN: Well done. Almost a yahtzee. JAMES CLEWITT: So now, I'm going
to throw three dice. And what we find is that we
get a similar plot, but it starts to curve over
like this. So we can go from 3 up
to [? 18 ?] now. What we're starting to produce
is a thing called a Gaussian distribution, a normal
distribution. It's called normal because
almost everything in nature does this. If there's some probability
distribution which is based on dice or based on the probability
of the behavior of lots of things which are
behaving identically, then you get a Gaussian distribution. So it's absolutely everywhere. And a Gaussian distribution
has a shape which looks a bit like this. And it's very sharply peaked. You're very, very likely to
get this average value. But it's still completely
random. I can't tell you what value
I'm going to get. All I can tell you is the
probability of that value. You don't want to hear about
the ancient Greeks, do you? We've thrown two. We've thrown three. Now, I've brought, frankly,
every die I own. And we're going to
throw many dice. And when we throw many, some of
them go on the floor, but what we end up with is this
Gaussian distribution. The probability of the number
that we're going to get is the Gaussian distribution. So what most people think of as
a random number is actually a uniform deviate, which
basically means that every number in the sequence has got
the same chance of coming up. But that doesn't
make it random. What makes it random is that you
can choose any number in the sequence. It doesn't matter what the
probability of that choice is. So you may be very, very likely
to choose the number in the middle. But that doesn't mean that it's
not randomly possible for you to choose any
number at all. BRADY HARAN: So it's like if you
and I had a weightlifting competition. The result will be random, but
you're probably going to win. JAMES CLEWITT: That's
not random. No, you misunderstand. I'm going to win that one. You've asked me to give
you a random number. And I really want to. So I've gone out of my way and
I've gone to the safe and I've got something that's going to
create a random number, because here, I've got a small
radioactive source. It's strontium-90. And it is going to randomly
give out electrons. As a neutron turns into a
proton, then out comes a high-energy electron
from the strontium. And so coming into the Geiger
counter here, I'm getting just as a steady stream
of electrons. And I'm counting them. And the number that I count-- surprise, surprise--
is Gaussian. But it's random. It's completely random. And so-- let's turn it on. Let me just show you. Can I show you? That's the happy sound
of radiation. So you can hear it, but
you can see it, too. Here it is. Here are the individual
pulses. Each one of these flashing
on here-- that's an electron going into
the Geiger counter, which is pretty cool. So I've written you're a
small computer program. I hope it works. The number you're going to
get is the deviation from the mean value. So I know what the mean value
for the count is going to be. And I'm giving you the
difference between that value and the number we
actually count. Here we go. [LAUGHING] Well, I was expecting it
to be a little more dramatic than that. What that means is we've got
exactly the mean value. So that is the most
likely outcome. And yet somehow I didn't
expect that that was going to be it. Shall we try again? We're generating another
random number. And to be honest, I really
want it to be 7. So this is it counting. 13-- it's prime. How much more random
do you want to be? That is a spectacularly
random number. It's 13. Brilliant. BRADY HARAN: So if you enjoyed
this video, I've got some good news for you. We're actually starting a new
channel a bit like Numberphile but all about computers. It's something a lot of
you have asked about. That channel's going to be
launching properly in May, but we've started it now
and you can go and have a look and subscribe. I'll put a link on the screen
underneath the video and in the video description, all
that sort of thing. Have a look, and we hope
to see you there, too.
Picked 7. Was annoyed. Won't pick 7 anymore in the future. Will still be annoyed.
07:50 "Everything in Nature follows a Gaussian distribution"
11:00 Measures a random process that follows a Poisson distribution.
At the start: "There is nothing random with single numbers. Only sequences can be random."
At the end: "I got 13. How much more random can you get?"
Picked 10. Nailed it.
So, does more dice make a game more or less random? How does one quantify randomness in terms of the probability distribution? Discuss.