41 Simulation

In this section we ask ‘what is normal?’

When we say ‘normal’ we’re talking strictly in the sense of numbers, not making a judgement about morals or values or any qualities whatever.  Nonetheless, it’s important to understand what we mean by normal in mathematics so that the judgements we do make based on numbers are grounded in solid foundations.

Let’s start with a problem.  Say you were pointed to a tree in early summer and asked to give your best estimate of how long a leaf is likely to be, if it were picked from the tree at random.  Another way of asking that question might be ‘what is the normal or average length of a leaf’?

Leaves

Come up with a method for finding the average length of leaves on an individual, fully grown tree.  Try to keep it plausible – that is, could you really do it within the space of an afternoon and without destroying the tree?

There is a huge array of different ways to do this.  However you did it, it’s likely that your method involved measuring a sample of (or all of) the leaves, writing down how long these individual leaves were and figuring out from this information what the ‘average’ length of the leaves were.  This would have involved some kind of calculation (to find the average leaf length, the calculation would be the sum of the length of all the leaves combined divided by the number of leaves in the sample, but there are other ways to do it too).

You might also have paid attention to the distribution of the leaves.  This might help us understand how likely we are to find a leaf of an average length.   What proportion of leaves were very short or very long compared to the rest of them?  We might well discover that, while we’re more likely to find an average length leaf than any other, they’re still fairly rare.  That leads us to another related concept: probability.

Probability

When the kind of distribution described above is graphed using the kinds of techniques we looked at in the section on interpreting visual information, it often creates the famous ‘bell curve’ shape.  This distribution does very well in describing a number of natural phenomena and random processes.  Understanding how this distribution works depends on a fundamental understanding of probability but also helps us understand some forms of randomness.  For example, in schools, understanding how the normal distribution works is essential for interpreting student results and statistics from standardised tests.

We’re feel comfortable and confident about asking the probability of a coin flip coming up heads or tails, or a rolling a 6 on a die.  But in truth, most of our intuition starts to get a little bit shaky once we start adding a little complexity.

Games of change

	What is the probability of a coin flipping sequence landing heads, then tails, then heads, then tails?
	How likely are you to roll 3 sixes in a row on a six-sided die?

These problems aren’t always easy, but thankfully, simulation can help.  On the next page we’ll look at the example of the Birthday Paradox to show you how.