Coincidences and the Lottery

Coincidences happen surprisingly often. Yet they are often not meaningful, i.e. they are “just a coincidence” and do not imply that we should change our worldview. For example, suppose there are a million people in contention for a lottery, and John Smith is found to win. Before knowing this, our probability for it is :

People often get afraid of this tiny probability, and proclaim something like “it’s not the probability of John Smith winning the lottery that is relevant, but the probability that someone wins”. However, this is anti-Bayesian nonsense. This tiny probability is, by Bayes’ rule, relevant for getting a posterior probability for . So how is it that we often still believe in the fair lottery (or that a coincidence is not meaningful)?

The answer is quite simple: the likelihood for the alternative, hypothesis, is just as small:
.
The reason is that before we knew who won, we had no reason to single out John Smith, and had to spread the total probability (1) over a million minus one alternatives (that the lottery was rigged in favor of one of the other entrants). Using analogous reasoning, yes, coincidences have tiny probability, but they also have tiny probability given the hypothesis of a mysterious force operating, because before the coincidence happened we didn’t know which of the multitude of coincidences were going to occur.

For more on this topic, you may be interested in this paper (by myself and Matt).

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Cricinfo inbox articles

For those who didn’t catch it: T. S. Trudgian has a nine-volume set of articles on Cricinfo purporting to demonstrate Why Australia can win the Ashes 5 – 0. Xavier Doherty a conspicuous omission.

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… Dust to Dust

Prediction time.

Australia 3, England 1

Man of the series: Mitchell Johnson

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Boltzmann, Philosopher

I have heard this quote from philosopher of science Paul Feyerabend before:

The withdrawal of philosophy into a “professional” shell of its own has had disastrous consequences. The younger generation of physicists, the Feynmans, the Schwingers, etc., may be very bright; they may be more intelligent than their predecessors, than Bohr, Einstein, Schrodinger, Boltzmann, Mach and so on. But they are uncivilized savages, they lack in philosophical depth — and this is the fault of the very same idea of professionalism which you are now defending.

I haven’t read much of the original works of “the predecessors” – most of my knowledge of their work comes filtered through textbooks and lecturers. I suspect this is true of most physicists. There are some good reasons for this – old papers aren’t easy to get a hold of, are awfully typeset, use archaic notation, and most of their content is purely of historical interest.

One exception is a paper I read recently by the great Ludwig Boltzmann, which is titled: On Certain Questions of the Theory of Gases. Its a three page Letter to Nature, a well-known publication which a lot of visitors to this site are trying to find when they Google its name. Here are a few quotes for you to mull over, to see if the “philosophical depth” comes through.

[Even if we had the ideal physical theory], we should still be a long way off what Faust’s famulus hoped to attain, viz, to know everything. But the difficulty of enumerating all the material points of the universe, and of determining the law of mutual force for each pair, would only be a quantitative one; nature would be a difficult problem, but not a mystery for the human mind.

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RIP Allan Sandage

Cosmic Variance has just informed me of the passing of Allan Sandage. Sandage is widely regarded as one of the great astronomers of our time. There is an anecdote in one of Timothy Ferris’ books (probably the Whole Shebang) of a group of astronomers who, with a flash new telescope at their disposal, decided to repeat some observations that Sandage had done many years before. They found, much to their amazement, that Sandage’s measurements were astonishingly accurate, especially given the instrument he was working with. He will be greatly missed.

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Conjecture of the evening

Especially for Cusp, I note the following (proof left for undergraduates):

(Convex h-index conjecture) For n chronologically distinct papers, each of which cites all previous papers, the corresponding h-index is the number of non-congruent diagonals in a regular polygon with number of sides 2 greater than n.

As a corollary, academics engaging in such cheeky behaviour may be indexed with the dimension of their corresponding polygon.

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What do you know? – A Fine-Tuned Critique of Ikeda and Jefferys (Part 2)

This is my second critique of the work of Ikeda and Jefferys (IJ) on the fine-tuning of the universe for intelligent life. IJ insist that we must always condition on everything that we know is true. Here, I’ll raise a few case studies in need of clarification. I should warn that I’m somewhat less certain about this part than the previous one. The fog is probably in my own head.

A. Magneto saves the day

This is a variation on John Leslie’s firing squad parable. You are sitting with your grandpa on his porch. Grandpa says, “I have a confession. I’m Magneto.” You: “What? You’re one of the Xmen? You can manipulate metals at will?” Grandpa: “Yes. That’s right”. You: “Right. Sure. Prove it.”

Grandpa pulls a set of keys from his pocket and makes them levitate two inches above his hand. “Yeah, nice magic trick, Grandpa”, you say. But then, up on the hill overlooking the porch, a freight train derails! Its carriages tumble toward the house. And, just your luck, this train happened to be loaded with TNT and samurai swords. The ensuing explosion sends several tonnes of rather pointy metal hurtling towards the porch. You instinctively flinch. A few seconds later … you’re alive! You turn in shock to see that every inch of your Grandpa’s house has shards of metal sticking out of it, except for two perfect silhouettes of you and your Grandpa. He looks at you, and smiles. “Not bad, huh?”

Now, like the nerd you are (you’re reading a science-themed blog, so there’s no point denying it), you want to formalise your conclusion.

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