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## How to win at the races

I’ve rambled about this before, but with the Melbourne Cup - “the race that stops a nation” – a few days away and Tom Waterhouse’s annoying face on TV too often, it’s worth repeating.

Don’t bet on the horse you think will win!

More precisely, don’t necessarily bet on the horse you think will win. Here is the only betting system that works:

1. For each horse in the race, and before you look at the price offered by the bookmaker, write what you think the probability (as a percentage) is that the horse will win. I.e. if the race was run 100 times, how many times would this horse win? You’ll have to do your homework on the field.
2. For each horse, take your probability and multiply it by the bookmakers price. Call that the magic number.
3. If any of the horses have a magic number greater than 100, bet on the horse with the highest magic number.
4. If none of the horses have a magic number greater than 100, don’t bet. Go home.

The magic number is how much (on average) you would make if you bet $1 on the horse 100 times, so it better be more than 100. The way that the bookmaker guarantees that they will make a profit in the long run is to ensure that no magic numbers are greater than 100. Because of the bookmakers slice (the overround), the odds are stacked against the average punter. You will only end up with a magic number greater than 100 if either you have made a mistake on step 1, or the bookmaker has made a mistake on his price. This leads to the following advice. You should only bet on a horse if a) You know more than the bookmaker, and b) The bookmaker has significantly underestimated one of the horses. Thus, the better the bookmaker, the more reason not to bet. And so, we come to Tom Waterhouse’s online betting business: “I’ve got four generations of betting knowledge in my blood. … Bet with me, and that knowledge can be yours.” This is exactly the information you need to conclude that you should never bet with Tom Waterhouse. The ad might as well say “bet with me; I know how to take your money”. You don’t want a bookmaker who knows horse racing inside-and-out, from horse racing stock, armed will all the facts, knowing all the right people. You don’t want a professional in a sharp suit surrounded by a analysts at computer screens. You want an idiot. You want someone who doesn’t know which end of the horse is the front, armed with a broken abacus and basing his prices on a combination of tea-leaf-reading, a lucky 8-ball and “the vibe“. You want a bookmaker that is going out of business. The more successful the bookmaker, the further you should stay away. The TAB was established in 1964, has over a million customers, 2,500 retail outlets, and made a profit of$534.8 million in 2011, up 14%. Translation: never bet with the TAB. Betfair’s profits were $600 million, SportingBet made$2 billion in 2009.  With those resources, they’ll always know more than you. If you’ve heard of them, don’t bet with them. Go home.

Hopefully you’re getting my point. Don’t bet on sports. If you go to the races, put on a nice outfit, drink a few beers and give the money to charity. If you must bet, have a random sweepstakes with your friends. You’ll get much better odds that way.

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## The Bayesian Utility of Derren Brown

I had the great pleasure a few nights ago to see Derren Brown‘s new illusionist / mentalist, hypnotist show Svengali. It’s fantastic, and highly recommended. If you’ve seen any of Derren’s previous shows on TV, then some of the routines will be familiar. This fails to make them any less baffling. If you’re unfamiliar with his work, here’s a sample:

(Here’s a bit more). One of the main themes of much of Brown’s work is his ability to recreate the “powers” of psychics, mind-readers and spiritualists without the pretence of supernatural intervention or paranormal activity. For example, in 2004 he performed a seance ”live” on channel 4, and in 2011 trained a member of the British public to become a faith healer.

There is an important and quite general lesson to be learned from Brown’s abilities. In the course of last night’s performance, Brown did a number of things which, if they had been performed by someone claiming psychic powers, would seem, if not totally convincing, at least on the way to suggesting psychic powers. I remain at a complete loss as to how Brown seems to read the minds of audience members and anticipate their seemingly free choices.

Suppose that Connie claims to be a witch – a real, proper, supernatural witch – and as proof of her powers, performs a great feat of mind-reading. Being the mathematical nerds that we are, we decide to formalise our inference that Connie is a witch (and should thus be burned). Help us, Rev. Bayes!

Let: (more…)

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## A mathematical puzzle (of sorts)

I happened across the following mathematical titbit, which I will set as a question to the reader. Let,

$\mu = (\sqrt{5} + 2)^{1/3}-(\sqrt{5} - 2)^{1/3}$      (1)

You are required to prove that

$\mu = 1$.

Striking, no? Obviously, sticking it into a calculator doesn’t count. I know a rather indirect way, which unfortunately involves the phrase “by inspection”. I’ll share it with you below the fold. There must be a nicer way! (more…)

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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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## P and NP are different, you say.

Just in time for the completion of the SUMS Puzzle Hunt 2010 (go Team Awesome!), & in memory of an earlier post of mine celebrating an attempted disproof of the Riemann hypothesis, here is a link to an attempted disproof of the equivalance of P and NP. I for one hope that Geraint will reprise his earlier effort at nonchalant disdain, though in the end I expect T. S. Trudgian, newly delivered to Lethbridge, to have the final say.

My next post will be less cliquey, I promise.

Update: Here is the only commentary I have come across so far that talks about the proof and possible objections in detail. Graeme also noted (in one of the ellipses below) that crowdsourcing was an excellent way to tackle evaluation of the proof. Little did I realise he had a concrete reference point in mind, but here is the P!=NP wiki.

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## What is Tuesday Boy telling us?

This has been killing me all week. It’s a probability problem known as the “Tuesday boy” problem. I’ll simplify the problem by reducing the possibility space.

Alice has two children. What is the probability that she has two boys given that:

a) at least one of her children is a boy?

b) at least one of her children is a boy; and at least one of her children is left handed?

c) at least one of her children is a boy, and he is left handed?

Assume that left/right handedness are equally likely.

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## Losing on a Winner

It was a feature of local news where I grew up that at the end of the sports report, a man in a tweed jacket would come on screen and recite a seemingly random sequence of names and numbers. I later worked out that he was the “tipster”, and that he was divulging his wisdom in the form of predictions as to who would win various horseraces the next day. Here is a more modern example.

I was thinking about this recently and realised something: every single horse racing “tip” is completely useless. Even if the tipster is almost completely certain that his tip will win the race, it doesn’t follow that you should bet on that horse.

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## Genus analogues

N.b. This is a technical post, written to illustrate a question I believe to be interesting to some colleagues outside my particular discipline. I am accutely aware of its shortcomings as expository work, and pedagogical criticism is almost as welcome as an attempt to engage with the question at hand.

(more…)

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## The other coincidence problem

J. S. Bloom points out:

$\Omega_{\Lambda,0} \approxeq 1 - \left(\frac{1}{\alpha_0}\right)^{1/e}/\pi^e;$

so putting constraints on the evolution of the fine structure constant away from its present value. With no apologies whatsoever to xkcd.

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