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Probability
Introduction
Probability Trees and Venn Diagrams
Conditional Probability
The Monty Hall Problem
The Birthday Problem
True Randomness

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ProbabilityTrue Randomness

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Most of this course relied on the fact that things like coins, or dice, or roulette wheels are completely random. However, that is not really true – we already learned that Edward Thorpe managed to predict the outcome of roulette.

Suppose we toss a coin: the chance of it landing heads is 0.5. If we knew which way the coin was facing just before it left the hand, we might be able to make a slightly better prediction, such as 0.58 or 0.41. If we also knew the weight and size of the coin, and the angle, position and speed as it left the hand, we could use the laws of physics – gravity, friction and air resistance – to model the motion of the coin and to predict the outcome. Finally, if we knew the exact position of every atom in the coin and of all the air molecules surrounding it, we could create a computer simulation to accurately predict what will happen.

One could argue that tossing a coin really isn’t random at all – it is chaotic. That means that the underlying physical principles are so complex that even tiny changes to the starting conditions (speed, angle) can have a dramatic effect on the final outcome. We can use coins in games and gambling not because they are random, but because it is so incredibly difficult (and for practical purposes impossible) to predict the result.

The same principle applies to many other “random” events in life, including dice and roulette wheels. They are not really random, we simply don’t have the tools to do the mathematical calculations accurately enough to predict the outcome.

But true randomness does exists – at the very foundations of matter. A block of radioactive material consists of billions of atoms which decay over time: they fall apart into smaller atoms while emitting dangerous radiation.

Physicists can calculate the probability that a particular atom will decay in a certain period of time, but it is impossible to work out which one will decay next – even if you know the exact properties of every atom.

The overall rate of decay, on the other hand, is so steady that it can be used to calculate the age of fossils that died thousands of years ago on Earth. This process is called Carbon dating.

Radioactive decay of atoms is caused by forces which act at much smaller scales within atoms, and which can be explained using Quantum mechanics. During the last century, physicists like Max Planck and Paul Dirac discovered that fundamental particles have a mind-blowing property: they can be in multiple different places at the same time. They don’t have a fixed position, but instead a probability distribution (also called wave function) which tells us how likely it is that we’ll find them at a particular position.

This incredible property is used by Quantum computers. Conventional computers can only ever do one computation at a time. Quantum computers can use the properties of subatomic particles to do many calculations at the same time – and that makes them significantly faster.

We can’t really understand or explain quantum mechanics – we just have to accept that it is what is predicted by mathematical theory and confirmed by physical observations. The curious quantum effects have only ever been observed on tiny scales of a few atoms, and it is not clear how they affect us in everyday life. But it is the only known effect in nature that produces true randomness.

Archie