## Flood Risk Puzzle

This is a ‘Natural Hazards problem’ that has been used to explore people’s understanding of risk. As usual I question the supposedly ‘mathematical’ answer.

Suppose that the probability that your house will be hit one or more times by the natural hazard during an exposure period of one year is .005. That is, if 1000 homes like yours were exposed to the natural hazard for one year, 5 of the homes would be damaged. Please estimate the probability that your home would avoid being hit by the natural hazard if exposed to the hazard for a period of 5/10/25/50 years.

You might like to ponder it for yourself.

In elementary probability theory the appropriate formula is 1-(1-p)n, where p=0.005 is the probability for one year and n is the number of years. But is this an appropriate calculation?

My in-laws were being charged a high property insurance premium because homes ‘like theirs’ were liable to flooding. They appealed, and are now paying a more modest premium. The problem is that a probability is rarely objective and individual, but experience-dependent on some classification. UK insurers rely on flood reports and surveys that are based on postcodes. Thus ‘properties like yours have a 0.5% risk of flood in any one year’ would really mean that some properties with your postcode have flooded, or have been assessed at being at risk from flooding. But, if like my in-laws your property is at the higher end of the postcode, this may not be at all appropriate. Just because the insurance company thinks that your risk of flooding is 0.5% does not mean that you should think the same.

Even from the insurance company’s perspective, the calculation is wrong. It would be correct if all post-codes were homogenous in terms of risk, but that clearly isn’t so. If a property hasn’t had a flood for 20 years then it is more likely to be at less risk (e.g., on higher ground) than those that have flooded. Hence its risk of flooding in the future is reduced. Taking more extreme figures, suppose that there are 2 houses on a river bank that flood every year and 8 on a hill that never flood. The chance of a house selected at random being flooded at some time in any long period is just 20%. And if you know that your house is an hill, then for you the probability may be 0%. In less extreme cases – typical of reality – the elementary formula also tends to overstate the risk. But the main point is that – contrary to the elementary theory – one shouldn’t just take probability estimates at face value. This could save you money!