The Dangers of Medical Statistics

Although modern cosmology requires a great deal of complicated statistical reasoning, I have it relatively easy because there is not much chance that any errors I make will harm anyone. Speculations about the anthropic principle or theories of everything are unlikely to be reported in the mass media. If they are, and are garbled, the resulting confusion is unlikely to be fatal. The same can not be said of the field of medical statistics. I can't resist the opportunity to include an example of...

Quantum Probability

So far I have focussed on what happens to single particles when quantum measurements are made. Although there seem to be subtle things going on, it is not really obvious that anything happening is very different from systems in which we simply lack the microscopic information needed to make a definition prediction. But quantum probability does have aspects that do not appear in classical stochastic processes. At the simplest level, the difference is that quantum mechanics gives us a theory for...

Random Processes

I have used the word 'random' quite freely so far without really giving a definition of what it means. The reason for that is that I really do not know. Turning the vague ideas we have about what it means into rigorous mathematical statements is surprisingly difficult. What I want to do in this chapter is look a little bit deeper into the concept, and look at how it applies (or does not) in both abstract mathematics and in physical systems. There are many different ways in which a sequence of...

Info

There is no correlation between the variables X and Y. their main-sequence lifetime because their cores have run out of hydrogen to burn. Giants are stars where nuclear burning is still going on, but either not in the core or not involving hydrogen. White dwarfs are not burning at all. They are collapsed objects so dense that they are governed by the laws of quantum mechanics. Their heat is not produced continually, but is trapped inside them by their incredible...

Intermezzo The First Digit Phenomenon

Before going on to talk about some of the more entertaining aspects of random behaviour, I thought it would be fun to introduce a quirky example of how sometimes things that really ought to be random turn out not to be. It is also an excuse to mention yet another astronomer. Simon Newcomb was born in 1835 in Nova Scotia (Canada). He had no formal education at all, but was a self-taught mathematician and Figure 13 Transition from laminar to turbulent flow in cigarette smoke. Reprinted with...

Cosmology by Numbers

It will be obvious that the Big Bang 'theory' is seriously incomplete. Because it falls apart at the beginning we have no way of setting the initial conditions that would allow us to obtain a unique solution to the system of equations. There is an infinitely large family of possible universes, so to identify which (if any) is correct we have to use observations rather than pure reason. The simplest way to do this is to re-write the Friedmann equation in a dimensionless form that is a bit easier...

The Logic of Uncertainty

The theory of probabilities is only common sense reduced to calculus. Pierre Simon, Marquis de Laplace, A Philosophical Essay on Since the subject of this book is probability, its meaning and its relevance for science and society, I am going to start in this chapter with a short explanation of how to go about the business of calculating probabilities for some simple examples. I realize that this is not going to be easy. I have from time to time been involved in teaching the laws of probability...

DNA Fingerprinting

I want to turn to a specific example of forensic statistics which has been involved in some high-profile cases and which demonstrates how careful probabilistic reasoning is needed to understand scientific evidence. Typically, the use of DNA evidence involves the comparison of two samples one from an unknown source (evidence, such as blood or semen, collected at the scene of a crime) and a known or reference sample, such as a blood or saliva sample from a suspect. If the DNA profiles obtained...

Probable Nature

The true logic of this world is the calculus of probabilities. This is a book about probability and its role in our understanding of the world around us. 'Probability' is used by many people in many different situations, often without much thought being given to what the word actually means. One of the reasons I wanted to write this book was to offer my own perspective on this issue, which may be peculiar because of my own background and prejudices, but which may nevertheless be of interest to...

What does it all Mean

The implications of quantum entanglement greatly troubled Einstein long before the EPR paradox. Indeed the interpretation of single-particle quantum measurement (which has no entanglement) was already troublesome. Just exactly how does the wave-function relate to the particle What can one really say about the state of the particle before a measurement is made What really happens when a wave-function collapses These questions take us into philosophical territory that I have set foot in already...

Are we Alone

Our Universe is certainly contrived in such a way as to make life possible within it. But just because it is possible, that does not mean that it is commonplace. Is life all around us, or did it only happen on Earth It fascinates me that this topic comes up so often in the question sessions that follow the public lectures I give on astronomy and cosmology. Do you think there is life on other worlds Are there alien civilizations more advanced than our own Have extraterrestrials visited Earth...

The Fermi Paradox

One day in 1950 the physicist Enrico Fermi went to lunch with colleagues from the Los Alamos National Laboratory in New Mexico, USA. Fermi had won the Nobel Prize for physics in 1938 for pioneering work he had done in nuclear physics in his home country of Italy until, dismayed by the rise of Fascism under Mussolini, he moved to the United States of America. In Chicago he had been responsible for the construction of the world's first controlled nuclear reactor (on a disused squash court). Over...

The Flatness Problem

I start with an illustration of how proper discussion of prior probabilities, within a Bayesian framework, can shed important light on fundamental issues connected with the behaviour of the classical Friedmann world models I discussed in the previous chapter. The cosmological 'flatness problem', as it is now known, arises from the peculiar behaviour of these models as they approach the initial singularity and needs to be addressed using careful dynamical arguments. Before doing this, however, I...