Archive for August, 2010

Intuition versus Calculation

August 31, 2010

WARNING: This post contains probability, which is not my strong point. If I have made mathematical or terminological errors, please correct them kindly in the comments.

Probability can be a bit of a minefield. I try not to rely on my intuition at all when someone asks me a probability question, because so often it can be wrong. So imagine my consternation on Friday evening at the end of a long hard day working with summer school pupils on probability when I was given a question by a close friend who should have known better, and asked to apply my intuition to it.

In order to pose his question, I’d better explain some terminology – skip this paragraph if you’re familiar with the idea of a uniform distribution. Imagine a number line with all the numbers between 0 and 1. You can pick any point on the number line you like. It doesn’t have to be a terminating decimal, it doesn’t even have to be a recurring decimal – it can be rational or irrational, close to 0, close to 1, or anywhere in between. The uniform distribution is a probability distribution where each of these choices is equally likely.

Start by imagining a random variable uniformly distributed on (0,1). It feels pretty intuitive that the expected value of such a random variable should be 0.5. My good friend’s question was as follows: What happens if you have two independent random variables, both uniformly distributed on (0,1), and you want to know the expected value of the smaller of the two variables?

My intuition said that the answer had to be less than 0.5, because for each random variable the expectation was 0.5, so for the two together and taking the smaller of the two, it just had to be less. My thinking then went very arm-wavy: some of the time they will both be less than a half, some of the time one will be less and one greater, and some of the time they will both be greater than a half, so maybe it’s about a third? My arm-wavy gut instinct was confirmed by my friend’s algebra.

I’m not too confident at manipulating probability density functions so I resorted to my usual tactic when scared off by continuous distributions: looked for a discrete analogue. What if I had a set of WHOLE numbers, and I picked pairs of them (with replacement, so the same number can come out twice)? I started with the set of numbers from 0-6 (deliberately picked a multiple of 3 in case thirds came into it). Actually that’s not entirely true; I started with the numbers from 1-6 and then changed my mind… This was a small enough set that I could list possibilities and calculate the average, and working on this special case gave me some insights to try working through some algebra for the set of numbers from 0-n. I leave this as an exercise to the reader 🙂

What I’d be most interested in though is if anyone has any insight to the original continuous version of the problem – how would you explain to a layman why the answer should be a third? Do you have a justification that doesn’t rely on integrating density functions and suchlike? Do you trust your instinct when it comes to probability, or are you cautious without calculation or experimentation to give you a feel for the problem?

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A week of proof

August 3, 2010

Last week we had some Year 10 students visiting NRICH as part of a summer school. The theme for the week was proof, and over the course of the week we worked on problems designed to expose the students to opportunities for convincing arguments, reasoning and proof.

I led a session about how to build up simple proofs. We started with these two problems (you might want to have a go for yourself before reading on):

Find three numbers whose squares add up to a multiple of 4.
What do you notice about the numbers?
Will it always be the case?
Can you prove it?

Find a pair of numbers whose squares add up to a multiple of 3.
What do you notice about the numbers?
Will it always be the case?
Can you prove it?
We discussed the first problem all together, starting by finding some examples and collecting them together on the board. Then we commented on what we noticed, and considered how to represent odd and even numbers algebraically to see if we could prove our conjecture that the three numbers had to be even. This led us into discussions of remainders when you divide by four, so we were effectively working on modular arithmetic although we weren’t using the notation. I pointed the students in the direction of the introductions to modular arithmetic on the NRICH site if they wanted to follow this up. Then they used what they’d learned from the first example to have a go at the second, and I was delighted that most of them quickly began to consider the remainders when they divided by three as being important.
I then threw in a quick one-liner to provoke some more thought:

Is 6n-1 a prime number for all positive integer values of n?
They weren’t having any of it – they found counter-examples extremely quickly, and this led to a couple of interesting discussions – that we need to have a body of watertight logic to prove something, but we only need one counter-example to disprove it. We also discussed the consequences for world mathematics if we HAD managed to find a formula that generated prime numbers…
We finished off with a couple of number patterns that lead to some nice algebraic proofs:

1x2x3 + 2 = 8
2x3x4 + 3 = 27
3x4x5 + 4 = 64
Will the answer always be a cube number?
1x2x3x4 + 1 = 25
2x3x4x5 +1 = 121
3x4x5x6 +1 = 361
Will the answer always be a square number?
This is the third study school these students have attended with us, and it’s a testament to how much they’ve progressed with their mathematical thinking that some of them dispatched the first problem almost immediately by representing the middle number as n and expanding some brackets. They found the second example a bit harder, until they were prompted to guess a form for the right hand side and expand left and right sides to see if they were equal.
This was just one session in a week devoted to thinking about what it means to prove something, and what level of justification is necessary – throughout the week I was delighted that they rose to every challenge we gave them, even when we asked them some pretty challenging questions. One session turned into a discussion of complex numbers, another touched on summing infinite series – ideas that Year 10 students might not normally be exposed to, but that they eagerly had a go at when it was presented to them. I hope they will go away from this study school with a better grasp of proof and a belief in themselves as young mathematicians who can persevere with hard problems.