## Tuesday, 25 June 2013

### 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.