NMSS 2012 – Strong Law of Large Numbers for a Coin Flip

The 2012 National Mathematics Summer School, held at Queens’ College affiliated to the University of Birmingham, and run by the United Kingdom Mathematics Trust, is drawing to a close today. I gave a problem-based talk on Probability to two groups of 20 junior students (15/16 year olds selected based on strong performance in national competitions for their agegroups), and a lecture to the six senior students (some of 2011’s strongest and most enthusiastic junior students) on the SLLN for the simplest non-trivial random variable imaginable: a coin flip.

In case any of the students, or indeed anyone else, is interested, a text of the problems, and the worked solutions that took up the majority of the lecture will be available here for a short while. Do email me if there are any questions!

Senior Probability Solutions. [Link removed. Email me if interested]

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Approaching Infinity

Last summer, I worked at and gave some lectures at the National Maths Summer School. The students submitted feedback forms, and a surprisingly large number mentioned that they would have liked to have a session about ‘infinity’. I was reminded of this by a post on an interesting blog that I’d seen linked to by, of all people, Stephen Fry. It is easy to forget, a full three years after a first university course on analysis, that the infinity had once seemed so confusing.

The problem is as much one of presentation as of mathematical content. The impression often given is that mathematical statements concerning infinity are not properly defined, or can’t be understood in a ‘real world’ setting. Unqualified and often rather misleading explanations are absolutely rife. And even some well-qualified scientists have put forward theories that are questionable at best. First we talk about some of the usual problems, and why they might not be so significant after all.

• No-one can imagine what infinity is: I’m not sure whether this is true – I personally feel I have a reasonable idea. But even this doesn’t matter. Arguments like this often reference the fact that there are atoms in the universe (or something similar) and how this doesn’t even compare to infinity. This is true, but it doesn’t affect our ability to understand and make deductions about a concept. I can’t imagine what 5-dimensional space looks like, but with five co-ordinates (x,y,z,w,v) I can describe it in mathematical terms that are entirely reasonable. This allows me to start working out properties of the object even if I can’t visualise it.
• Infinity is about philosophy: This might well stem from its appearance in popular culture (‘to infinity and beyond’) and the metaphysical (‘the Father of an infinite majesty’ etc). I would suggest that if you are worried about coming to a philosophical understanding of infinity, first you should question whether you have a true philosophical understanding of seven. I can picture seven oranges in my mind, but does that alone really explain all the seven-ness of seven? In any case, we can learn some simple rules to deal with seven (like 3+4=7) in a concrete way, and though the rules aren’t as ‘obvious’, we can do the same for infinity.
• Infinity is not a number: Again, this is in some sense true (see below). But it doesn’t make any difference if you use it correctly. At various points in time 0 has been considered ‘not a number’, as have negative numbers. If you build up the world of complex numbers by defining the square root of -1, is this a number? As with many words, infinity means different things in different contexts. This is actually often really about the following: