It rains today. The coin lands on heads. I don’t have an accident. Who
chose these outcomes or were they predetermined? If they were predetermined,
why do we think of these as random events? This isn’t an article about
probabilities but about how randomness occurs, or seems to occur in our
everyday lives.
1 Coin
Flips
Let’s looks at a
coin flip. Our thumb pushes against the coin into the air causing it to turn
over and over. The amount of effort of the thumb, the trajectory of the coin
and to lesser extent the pressure and resistance of the air itself control how
this coin flips. Eventually the coin hits the ground and depending on the angle
will settle into one of two low-energy states, either with the “heads” side of
top or the “tails” side.
There is nothing particularly random about this process. Every aspect could
be controlled and simulated. Whether the coin lands on “heads” or “tails” is
basically determined as soon as the coin leaves the thumb. Nevertheless, we
allow or often require that one player call “heads” or “tails” while the coin
is in the air and still treat whether the coin lands on “heads” or “tails” as a
random event.
The weather and my safe driving depend on far more complex chains of events
though again they seem predetermined from the base conditions. Two questions
stick out:
-Why do we consider this processes random?
-Is there true randomness in nature?
We’ll tackle the
second question first.
2 Randomness
in Nature
“God does not play
dice” famously claimed Albert Einstein, a believer in scientific determinism.
Before the 20th century many scientists believed the same, that the world and
the universe moves following a path full defined from its current state. In 46
the 20th century we saw the development of quantum mechanics that made us question
this philosophy.
Suppose you have a light bulb and put in front of that bulb a piece of
cardboard with a thin vertical slit. Then through that slip light will come
through only oriented in a vertical direction. We can check that easy enough by
putting another cardboard with another slit in front. If that second cardboard
has a vertical slit lined up with the first cardboard, then all the light
passing through the first cardboard passes through the second. If the second
cardboard has its slit oriented horizontally then no light will go through
What if we orient the second cardboard’s slit at a 45-degree angle? About
half of the light will go through. As we reduce the light coming from the
source, we equally scale down the light that comes through the 45-degree slit.
According to quantum mechanics, light is not made up of a substance that
one can make arbitrarily small. Instead light comes in packets, or quanta.
Think grains of sand that make up a beach. We can reduce the light source so it
emits a single quanta of light, a photon. What happens if that vertically
oriented photon hits the 45-degree slit?
You can actually run this experiment and put a photo detector at the other
end of the cardboard with a bell that will ring if a photon is detected.
Perhaps not surprisingly, half of the time the photon is blocked and the other
half of the time it comes out of the slit oriented at the 45-degree angle. Half
of the time the bell is rung.
This seems like true randomness, a completely controlled and reproducible
experiment that has two distinct outcomes, a bell ringing or a silent bell,
each occurring seemingly independent and with probability one half. God does
seem to roll dice to choose whether or not to ring the bell.
Or maybe not? Perhaps we are just observing a piece of a larger
deterministic system and by measuring the photon we reduce the dimension of the
system we see but it is still part of the larger picture. This is a bit
confusing so let’s consider what happens if we don’t observe the outcome.
Suppose instead of ringing a bell, the detector releases a poisonous gas into a
box containing a live cat. If a photon is detected the cat is killed, otherwise
that cat continues living unaware of the unreleased gas. Suppose we don’t look
inside the box. Thus is the tale of Schrödinger's cat.
Without looking inside the box we don’t know whether the cat is alive or
dead, whether or not the photon was detected. We can think of either that the
cat is alive or dead and we just don’t know the answer. Or we can view it as a
quantum state, where the cat is both possible alive or dead in quantum
superposition. Only when we open the box and view the cat does the
superposition collapse into one of the states where either the cat is alive or
dead. Similarly, someone sitting outside our universe can model out universe
with a deterministic transition of quantum states in superposition as long as
they never look inside the system.
So do we get true randomness or not in quantum mechanics? There is no true
clear answer and is more a philosophical debate than a scientific one. There
are other potential sources of randomness, for example black holes that seem to
destroy information, that are even harder to reason about.
While physics doesn’t yet give us a clear answer of whether we have true randomness in nature, this question does not address the flipping coin. No single photon knocks this coin off its course. The coin flips in highly controlled experiment will always land the same way. How can randomness happen when events are determined?
3 Randomness
from Complexity
When we flip a coin, a very deterministic process, why do we consider whether the coin lands on heads or tails random? With the proper sensors and enough computing power we could determine the coin’s outcome from its movement in the air. Typically, the coin we flip these coins between two humans and we just lack the computation power to compute the answer. The outcome of the coin is unpredictable to us, and being unpredictable we treat the answer as a random event. Our inability to compute the answer makes the outcome for all practical purposes random to us.
We can say the same for the weather. The weather agencies have powerful
tools and computers to predict the weather but they have to use limited models
because even the most powerful machines cannot take into account all the
factors that may drive the weather, even for the next day. A weather forecaster
still gives a percentage of rain treating rain like a randomized event.
In a casino, the dice and roulette wheels are simple devices yet rely on a
number of complex interactions that they too seem completely random. Random
enough that casinos literally put their money on the line on the assumption
that the bettors cannot predict the outcome of a dice throw or a roulette wheel
better than random chance. In craps many casinos will allow the bettors
themselves to throw the dice, knowing that even doing so won’t give them an
advantage in predicting the outcome. In blackjack, the dealer will often
shuffle the cards directly in front of the bettors, and yet the bettors cannot
treat the deck as anything but in a perfectly random order. Even card counters
assume the deck is perfectly random, they just use the outcomes of earlier
cards to adjust the probabilities of later ones.
In financial markets traders need to assume a probability on future prices
in order to properly price securities because these prices depend on a complex
way on a series of events and future trading.
In the United Kingdom, you can bet on the outcome of sporting events,
elections, winners of award ceremonies. The betting sites don’t take a risk
here, they set betting odds so that both sides are equally represented and they
make money independent of the outcome. But the bettors must make probability
estimates, consciously or unconsciously, in order to decide on which outcome to
make the bet.
Even consider the game of chess. There is no obvious randomness in chess.
The current board position is fully known to both players and no random
devices, such as dice used in backgammon or card shuffling used in poker, are
involved in a chess match. Yet we talk about risky moves and how likely we
think white will win after a certain move in the game. The complexity of chess
turns this game of “perfect information” into a game of “imperfect information”
seemingly adding a measure of randomness to a game with no obvious source of
randomness.
When you ask a computer for a random number, it doesn’t truly give you a
random number. Rather it gives you the result of a complex calculation, to give
you a number you can treat as random. Similarly, cryptographic protocols make
real messages appear to be truly random to those who don’t have the proper
decryption keys. There are a number of theoretical results that show how to
convert any sufficiently hard function into a pseudorandom generators and
cryptographic protocols that cannot be distinguished from true randomness. In
practice we have also developed protocols that no man or machine can
distinguish from purely random.
4 Removing
randomness by beating complexity
With better and
more powerful algorithms and computers we can sometimes beat randomness built
from unpredictability and complexity. Our recent ability to access large
amounts of data combined with machine learning algorithms now made feasible on
current technology can help break through the unpredictability barrier. We
cannot usually completely predict with confidence, but we can gain enough
information about future probabilities to get an advantage over those who still
view the original events as completely unpredictable.
New models, stronger computers and better algorithms have greatly improved
weather forecasting though we are still a long way from forecasting with
certainty. Hedge funds use deep mathematical techniques to gain an advantage in
trading securities. Some sophisticated bettors can find small imperfections in
roulette wheels using hidden computation devices and use that to gain a small
but real advantage in betting. A chess playing computer, even housed in your
smart phone, can beat any human these days by doing a better job of predicting
the chances of winning from potential future game boards better than humans
can.
5 What is
randomness?
Whether we get true
randomness from nature depends on the interpretation of what nature does. Truly
what we view as randomness is not randomness at all, rather it is simply what
we cannot predict given the complex processes that generates the outcomes of
events.
