Probability chapter rescued

I am delighted to day that at the last minute my publisher has agreed that we should reinstate the chapter on Probability in the 5th edition of Mathematics Explained for Primary Teachers (see my post on this site on 23 December). We were alerted to the fact that the enlightened Welsh and Scottish primary curriculums still include probability - and the book serves the other UK countries as well as England. And, of course, there are plenty of academies and free schools who do not have to follow the National Curriculum (ironic, isn't it!), so we want to make sure that teachers in these schools have the opportunity to understand this most important and interesting application of mathematics and to consider exploring it with the children they teach.

Now here's an intriguing use of numbers that turned up at home recently. I said to Mrs H, 'What's four plus one?' She replied, 'Thirteen'. She was right, of course. Can you explain?

And, finally, just to grab a little reflected glory: the amazing Jon Haylock doing astounding things around the Welsh coast with Griff Rhys-Jones on ITV this evening (8 pm) is my nephew.

Probability and breast cancer

The newspaper headlines today provide yet another example to support what I posted yesterday about how the concepts of probability and risk have become central to our decision-making. The National Institute for Health and Care Excellence (NIHCE) has been reported today as saying that tamoxifen or raloxifene taken daily for five years can cut breast cancer risk in women by 40%.

Now the risk of contracting breast cancer is itself always expressed as a percentage. For example, across the entire population of women in the UK the current probability of a woman contracting breast cancer sometime in her life is given as 12%. This means that, on the evidence of current statistical data, it is estimated that on average about 12 out of 100 women chosen at random from the entire population will develop breast cancer.

Now within that population there are subsets of women who, because of genetic and other factors, have a higher or lower probabilities than this 12%. So, consider an example of a woman for whom the risk of getting breast cancer in her lifetime is calculated as 50%. What does it mean for the NIHCE to say that the risk is cut by 40% if she follows a particular medication regime? Sadly, it does not mean that the risk is reduced to 10%.

There is always a difficulty in understanding statements about probability that are based on percentages of percentages. This has been made clear by some of the comments made on today's report. In this example, to reduce a risk of 50% by 40% reduces the risk to 60% of 50%, which is 30%. So, in this example, the woman taking the prescribed medication has her risk of contracting breast cancer reduced from 50% to 30%. That's worth doing, of course, but she still lives with a higher-than-average risk.

So, I repeat my argument: that understanding probability is so central to real-life decision-making that the sooner we start getting children to understand the basic concepts of probability and risk the better. 

There's research evidence (Schlottman, 2001) that children as young as 6 years can intuitively understand the idea of risk and can simultaneously take into account both the likelihood of an outcome and the reward or penalty associated with it. So, it will be a real pity if teachers cannot build on this intuitive understanding of functional probability within the primary school mathematics curriculum through learning experiences that help children to construct a better understanding of such a hugely important topic.

Schlottman, A. (2001) 'Children's probability intuitions: understanding the expected value of complex gambles', Child Development, 72(1): 103–22.

Probability a great loss

Michael Gove's new primary curriculum proposals for mathematics have removed 'probability' from the programmes of study, presumably to make room for lots of extra hard, abstract and pointless calculations. What a shame! Probability is a really important application of mathematics that is fundamental to all kinds of discourse: political, medical, sociological, educational, and so on, and so on. And it's also great fun to introduce to children and helps them to make sense of a world where decisions are based continually on assessment of risk.

It is also an area of mathematics that people get wrong. Today's example was provided by Jack Straw, the former Home Secretary, commenting on the Stephen Lawrence enquiry on Radio 4. I think he meant to say that it is very unlikely that anything like the actions of the police 20 years ago could happen now. In fact, what he said was: 'the chances of that happening today are infinitesimally smaller than they were 20 years ago.' So, no change then!

Friday operations

More misrepresentation in reporting statistics in the news media today! This time about the likelihood of dying after surgery at the end of the week.

I have just sent this letter to The Times:

To report that the 'risk of dying after operation soars towards the end of the week' (The Times, 29 May) is to distort the statistics that the  probability of dying after a Friday operation is 0.82% compared with 0.55% for a Monday. 

These results could be reported by saying that a patient has 99.18% and 99.45% chances of surviving Friday and Monday operations respectively. Both are extremely high chances and the survival rate for Mondays is only slightly better than that for Fridays. If I needed surgery I would be grateful to take either.

Mathematical Problem 2: Solution

The problem (set on 26 March 2010) was this:

You choose eight different books randomly from a library shelf. If you turn to the same numbered page in each book (for example, page 5), how likely is it that on at least two of these pages the text will start with the same letter?

The correct response from the options is ‘highly likely’. I tried this experimentally. With 50 trials I recorded 48 Yes results. So my estimate of the probability of at least two pages starting with the same letter is 0.96! To me that seems counter-intuitively high.

This is the best way of approaching this problem – doing lots of trials and using the relative frequency of ‘Yes’ as an estimate of probability.

You can approach it theoretically – but to make it accessible we have to assume that each of the 26 letters is equally likely. On this basis the probability of the second book producing a different letter from the first is 25/26.

Then the probability of the third book producing a different letter from the first two is 24/26. And the fourth book being different from the first three is 23/26. And so on … until we get a probability of 19/26 for the eighth book being different from the first seven looked at.

For all the books to be different you need all these things to happen, so you multiply together the probabilities: 25/26 X 24/26 X 23/26 X 22/26 X 21/26 X 20/26 X 19/26, which equals approximately 0.3. So, the probability of them not being all different is 0.7.

Why is this so different from my experimental result? This is because of the assumption that all letters are equally likely. This is clearly not the case. In my trials I did not have any examples of the first letter being x or z (I didn’t use algebra books!). And certain letters, such as t, a and s turned up much more frequently than 1 in 26. This increases substantially the probability of letters being duplicated.