Archive for the mathematics Category

Joseph Bertrand and the Monty Hall Problem

Posted in Bad Statistics, History, mathematics with tags , , , , on October 4, 2017 by telescoper

The death a few days ago of Monty Hall reminded me of something I was going to write about the Monty Hall Problem, as it did with another blogger I follow, namely that (unsrurprisingly) Stigler’s Law of Eponymy applies to this problem.

The earliest version of the problem now called the Monty Hall Problem dates from a book, first published in 1889, called Calcul des probabilités written by Joseph Bertrand. It’s a very interesting book, containing much of specific interest to astronomers as well as general things for other scientists. Ypu can read it all online here, if you can read French.

As it happens, I have a copy of the book and here is the relevant problem. If you click on the image it should be legible.

It’s actually Problem 2 of Chapter 1, suggesting that it’s one of the easier, introductory questions. Interesting that it has endured so long, even if it has evolved slightly!

I won’t attempt a full translation into English, but the problem is worth describing as it is actually more interesting than the Monty Hall Problem (with the three doors). In the Bertrand version there are three apparently identical boxes (coffrets) each of which has two drawers (tiroirs). In each drawer of each box there is a medal. In the first box there are two gold medals. The second box contains two silver medals. The third box contains one gold and one silver.

The boxes are shuffled, and you pick a box `at random’ and open one drawer `randomly chosen’ from the two. What is the probability that the other drawer of the same box contains a medal that differs from the first?

Now the probability that you select a box with two different medals in the first place is just 1/3, as it has to be the third box: the other two contain identical medals.

However, once you open one drawer and find (say) a silver medal then the probability of the other one being different (i.e. gold) changes because the knowledge gained by opening the drawer eliminates (in this case) the possibility that you selected the first box (which has only gold medals in it). The probability of the two medals being different is therefore 1/2.

That’s a very rough translation of the part of Bertrand’s discussion on the first page. I leave it as an exercise for the reader to translate the second part!

I just remembered that this is actually the same as the three-card problem I posted about here.

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R.I.P. Maryam Mirzakhani (1977-2017)

Posted in mathematics with tags , , on July 18, 2017 by telescoper

Very sad news arrived at the weekend of the death of the brilliant Iranian-born mathematician Maryam Mirzakhani of breast cancer at the age of just 40. Let me first of all express my heartfelt condolences to her family, friends and colleagues on this devastating loss.

A uniquely creative and inspirational figure, Maryam Mirzakhani was the first woman ever to win the coveted Fields Medal; her citation for that award picks out her work on the dynamics and geometry of Riemann surfaces and their moduli spaces.  Here’s a short video of her talking about her life and work. It’s fascinating not only because of the work itself, but the insight it gives into the way she did it – using very large sheets of paper covered in drawings and notes!

R.I.P. Maryam Mirzakhani (1977-2017).

 

Betting on the Supreme Court

Posted in mathematics with tags , , , , , , , on December 6, 2016 by telescoper

This week the UK Supreme Court is hearing an appeal by HM Government against the judgment recently delivered by the High Court which was that the UK Government must seek the approval of Parliament before it can invoke Article 50 of the Lisbon Treaty and thus begin the process of leaving the European Union. You can watch the proceedings live here. I had a brief look myself this morning but as I’m not a legal expert I found it rather hard to follow as it’s rather technical stuff. That wasn’t helped by the rather dull delivery of James Eade QC who was presenting the government’s case. Nevertheless, it is a very good thing that we can see how the law work in practice. I was surprised at the lack of gowns and wigs!

Although Eade seemed (to me) be on a very sticky wicket for some of the time, it’s impossible for me to come to any informed inference about who’s likely to win. Out of interest, to see what other people think, I therefore had a quick look at the betting markets. Traditional bookmakers (such as William Hill) are offering 1-3 (i.e. 3-1 ON) for the original decision being upheld so they’re clearly expecting the appeal to fail.

These days, however, I’ve started to get interested in other kinds of betting markets, especially the BetFair Exchange. This allows customers to act as bookmakers as well as punters by offering the option to “lay” and/or  “back” various possible bets. “Laying” betting means effectively acting as a bookie, proposing odds on a particular outcome. i.e. selling a bet.  “Backing” a bet means buying a bet. The exchange then advertises this to prospective bettors who sign up of they are prepared to stake money on that particular outcome at those particular odds. It’s very similar in concept to other trading services, e.g. share dealing. Matches aren’t always made of course, so not every bet that’s offered gets accepted. If that happens you can try again with more generous odds.

The advantage of this type of betting is that it represents an “efficient market”. Such a market occurs when all the money going into the market equals all the money being paid out in the market – there is no leakage or profits being taken. Efficient betting markets rarely exist outside of betting exchanges – bookmakers need to reap a profit in order to run a business. For example, though William Hill is offering 1-3 on the Supreme Court ruling being upheld, the odds they offer against this outcome are 12-5. These are not “true odds” in the sense that they can’t represent a consistent pair of probabilities of the two outcomes (as they don’t add up to one). In the case of an exchange market a bet laid at 1-3 is automatically backed at 3-1. These can then be regarded as “true odds”.

This is what the BetFair Exchange on the Supreme Court hearing looks like at the moment (you might want to click on the image to make it clearer):

 

supreme-court

The odds are given in a slightly funny way, giving the gross return for a unit stake (including the stake). In more normal language “4.3” would be 100-30, i.e. a £1 bet gets you £3.33 plus your £1 back. A bet on “overrule” at “4” (3-1) corresponds to a bet against “uphold” at 1.33 (1-3), reflecting what I was saying about “true odds”.

The first thing that struck me is the figure at the top right: £38,427. This is the value of all bets matched in this market. By BetFair standards this is very low. A typical Premiership football match will involve bets at least ten times as big as this. As in the court case itself there just isn’t very much action!

Apart from that you can see that the odds here are broadly similar with William Hill etc with implied odds around 3-1 to 4-1 against overruling.

Before you ask, I’m not going to bet on this myself. My betting strategy usually involves betting on the outcome I don’t want to happen. Although I think Parliament should be involved in Article 50 I am just happy that this matter should be left to our independent judiciary to decide.

Romanesco and the Golden Spiral

Posted in mathematics, The Universe and Stuff with tags , , , on November 8, 2016 by telescoper

Some time ago I mentioned that I received one of these in my weekly veggie box..

romanesco

Actually, that reminds me that a new box is due tomorrow morning…

Anyway, the vegetable in the picture is called Romanesco. I’ve always thought of it as a cauliflower but I’ve more recently learned that it’s more closely related to broccoli. It doesn’t really matter because both broccoli and cauliflower are forms of brassica, which term also covers things like cabbages, kale and spinach. All are very high in vitamins and are also very tasty if cooked appropriately. Incidentally, the leaves of broccoli and cauliflower are perfectly edible (as are those of Romanesco) like those of cabbage, it’s just that we’re more used to eating the flower (or at least the bud).

It turns out that this week’s Physics World has a short piece on Romanesco, which points out that a “head” of Romanesco has a form of self-similarity, in that each floret is a smaller version of the whole bud and also displays structures that are smaller versions of itself. That fractal behaviour is immediately obvious if you take a close look. Here’s a blow-up so you can see more clearly:
romanesco-broccoli2-550x412

However, one thing that I hadn’t noticed before is that there is another remarkable aspect to the pattern of florets, in that they form an almost perfect golden spiral. This is a form of logarithmic spiral that grows every quarter-turn by a factor of the golden ratio:

\phi = \frac{1+\sqrt{5}}{2}.

Logarithmic, or at least approximately logarithmic, spirals occur naturally in a number of settings. Examples include spiral galaxies, various forms of shell, such as that of the nautilus and in the phenomenon of phyllotaxis in plant growth (of which Romanesco is a special case). It would seem that the reason for the occurrence of logarithmic spirals  in living creatures is that such a shape allows them to grow without any change in shape.

Not really relevant to anything much, I know, but I thought you might be interested…

P.S. One thing the Physics World piece fails to mention is that, regardless of its geometrical properties, Romanesco is really delicious!

Computable Numbers, 80 Years on..

Posted in History, mathematics, Uncategorized with tags , , , , on May 28, 2016 by telescoper

There’s been rather a lot of sad news conveyed via this blog recently, so I thought that today I’d mark a happier event. Eighty years ago today (i.e. on 28th May 1936), a paper by Alan Turing arrived at the London Mathematical Society. Entitled “On Computable Numbers, with an Application to the Enstscheidungsproblem“, this was not only enormously influential but also a truly beautiful piece of work. Turing was only 23 when he wrote it. It was delivered to the London Mathematical Society about 6 months after it was submitted,  i.e. in November 1936..

Here’s the first page:

Turing

The full reference is

Proc. London Math. Soc. (1937) s2-42 (1): 230-265. doi: 10.1112/plms/s2-42.1.230

You can find the full paper here. I heartily recommend reading it, it’s wonderful.