Let us play a game. You give me 10 euros. We toss a coin. If it is heads I return you 30 euros and you win 20 euros. If it is tails you loose and I keep your 10 euros. Excellent deal, is it not? From a statistical point of view it definitely is, because the expected net return is positive: 0.5 × 20 – 0.5 × 10 = 5 euros. I am sure you will be happy to take on this bet. Now let us raise the stakes. You give me 100 euros, we toss a coin, if it is heads you get 300 euros back, if it is tails you loose your 100 euros. Will you still play? And what if we raise the bet to 1,000 or 10,000 euros? If your name is Bill Gates you might still play but at some point (10 billion euros?) even Bill will have to stop because he simply cannot afford the risk anymore. At that point the consequences of losing become too serious and are no longer outweighed by the benefits of winning. The option with less uncertainty (don’t play!) is preferred, even though it has a lower expected net return. This tells us that people tend to be risk-averse (https://en.wikipedia.org/wiki/Risk_aversion).
The above illustrates how important it is to quantify uncertainties and probabilities, because risk calculations can only be done if the probabilities are known. For instance, we all know that it would be wrong to design an urban drainage system on just the average rainfall. Instead, the system design should be based on the full rainfall probability distribution, because we want to make sure that the probability of a sewer overflow is small. We must be prepared for extreme events because the financial and environmental costs of an overflow are simply too high. One of the main aims of the QUICS project (https://www.sheffield.ac.uk/quics) is to quantify the uncertainties associated with urban hydrologic models and their inputs, and propagate these to model outputs. It provides crucial information for risk-averse decision making.
Whenever I suggest the coin toss game to my students (only hypothetically, of course) they usually go along but not for very long, because they have a tight budget. I was never really offered the game myself, but being a mathematical engineer and reasoning rationally I think I would go as far as 2,000 or perhaps even 5,000 euro. However, recently I learnt that if it were for real I probably would not. A few weeks ago I was at a Risk Analysis meeting of the International Union of Forest Research Organizations (http://riskanalysis-iufro.org//meetinginformation.html). Experts in risk perception present at the meeting explained me that the human brain functions such that the ‘pain’ caused by a unit loss is felt 2.5 times stronger than the ‘joy’ felt by a unit gain. If this were true, nobody would enter the coin toss game. Whether this also implies that nobody would ever buy a lottery ticket I do not know. I do know that many people do buy such ticket, even though they are aware that the expected net return is negative. Intriguing, is it not?
We can also play the coin toss game in a slightly different way: you decide how much money you put at stake (be it 0, 1, 5, 200 or 10,000 euro). We play the game with your bet of K euros. You have a 50% chance of losing the K euros and a 50% chance of winning 2K euros. How large is your K?