The Monty Hall problem, etc

I just came across this problem again in my web surfing - this is one of the many links that Google gives that provides an explanation of the problem and the counter-intuitive solution. I must admit it is making my head hurt a little bit.

So, now I should go back to somehow trying to algebraically solve these two questions from my latest tutorial exercise:

1. 2*tan x + cos x = 3 for x between 0 and 360 degrees

2. (sin x)^2 + tan x = (cos x)^2

I've solved them using graphing software, but we're supposed to be able to solve them algebraically, and I just keep getting in a huge tangle and ending up with cubics or quartics which don't give me the right solutions. Either I'm doing something dumb, missing something easy, or there's a misprint in the questions - unfortunately my tutor isn't available till Tuesday, so meantime I keep going back to them and trying again. Hints always welcome.

Science, philosophy and religion

Yesterday, while surfing the net, I found this book called “Galileo, Darwin and Hawking: The Interplay of Science, Reason and Religion” by Phil Dowe – now at the University of Queensland (UQ). This is the publicity blurb for it from Dove Booksellers :

The history of the interaction of science and religion is fraught with tension rather than harmony. Only recently have philosophers like Phil Dowe begun to seriously relate these two pillars of human civilization. This fascinating book discusses with insight and verve the relationship between science, reason, and religion, giving special attention to the most conflicted topics - cosmology, evolution, and miracles.

Providing a concise introduction to the philosophy of science suitable for all readers, Dowe's “Galileo, Darwin, and Hawking” shows that there are basically four ways to relate science and religion. Two of them, "naturalism" and "religious science," present these endeavours as antagonistic. By contrast, the "independence view" understands them as wholly unrelated. Finally, the "interaction view" sees religion and science as complementary. Dowe defends this last perspective as most truthful and helpful to our time, arguing his case by exploring the history of science, highlighting the life and work of three giants of science: Galileo Galilei, Charles Darwin, and Stephen Hawking.

I’ve added it to my “must buy” list – this is exactly the sort of “interplay” I’m interested in. Part of the reason I want to study physics is so that I understand enough about it to start to ponder what it all means. I’m mindful of the admonition from John Baez in his “How to learn maths and physics”:

Warning: there's no way to understand the interpretation of quantum mechanics without also being able to solve quantum mechanics problems - to understand the theory, you need to be able to use it (and vice versa). If you don't heed this advice, you'll fall prey to all sorts of nonsense that's floating around out there.

Whilst I am very much attracted to the idea that modern physics is starting to confirm some very old spiritual ideas – eg everyone and everything is connected – I really don’t want to take such ideas on board unless there is a solid basis for them – and if that means I have to understand quantum mechanics in order to evaluate these ideas, then that is one of my aims.

Now if only I was healthy enough to actually attend UQ and study physics, philosophy and religion properly.

Maths - exam number one

On Saturday, I sat the first of two exams for the maths subject I’m studying (effectively the exam was a year 11 one). It went well – I finished nearly an hour early (it was a three hour exam) and decided to leave so I didn’t “fix” an answer that didn’t need fixing – I’d already checked everything twice.

I found it quite difficult to prepare for, though – my usual exam preparation technique consists of doing revision notes, re-working any problems that I had noted on the way through as being particularly difficult, memorising notes to the best of my ability, then doing a series of past exam papers as practice exams under exam conditions. This gives me a good idea of where my weak points are (the things I need to cram into my head just before I walk in and get down on paper the minute I’m allowed to write) and also some idea of whether the time provided for the exam is likely to be enough for me to do it in.

This time, there were no previous exam papers available – not only that, but there was no information on what percentage of the overall marks the exam contributes – nor, when I got the exam, was there any indication of what marks each individual question was worth.

Now, normally when I first get into an exam room, and I’m allowed to pick up a writing implement (and after writing down those easily forgettable things I mentioned earlier), I work out roughly how many minutes per mark I have and draw up a quick list of how long that translates to for each question – that way I find I’m not spending way too much time on a question that is not worth many marks – I also usually try to allow for 10 minutes review time at the end to find and fix the inevitable “stupid” mistakes that I always seem to make (like the 2 + 2 = 5 that slipped through the net in my final year at high school).

So I found myself constantly checking the time as I finished each question and wondering whether I was on track or not – I was continually fighting the temptation to just rush through questions, because until the first hour or so had elapsed, I really didn’t know whether I would finish in the allotted time or not.

Anyway, hopefully I passed, and if I can do exams under these conditions, maybe I can manage university exams – which is one of the questions this studying is meant to answer for me. My next maths exam will probably be in a few months time once I get through the rest of the year 12 book – at the moment I’m a bit stuck on some questions which require drawing graphs by hand – I can’t help wondering why I should bother to learn to do this by hand when I would assume that nowadays graphs are drawn using computer software.

Still, the sooner I get through the maths, the sooner I can enrol in physics, then chemistry, then biology – and then I can seriously look at university options – still around 2 years away, methinks.