Module: Basic statistics
Quote of the page
If you have knowledge, let others light their candle by it.
- Margaret Fuller
Most risks don't involve money--at least, not directly. For example, if you are worrying about the possible side-effects of a particular medication, your worry is not primarily about money. If we are going to be able to apply the concept of expected value to such contexts, we will need a way of comparing outcomes in terms which are not financial.
The way to do this is via the concept of utility. The utility of an outcome provides a numerical measure of how good or bad the outcome is. To take a trivial example, suppose I am trying to decide whether to take my umbrella with me today. There are four possible outcomes, because it may or may not rain, and I may or may not take my umbrella. If it rains, it is clearly better to take my umbrella than not, but if the weather is fine, then it is marginally better not to take my umbrella, so that I have fewer things to carry. The best outcome for me is if it's fine and I don't take my umbrella, so I'll assign that outcome a utility of +10. The outcome in which it's fine and I do take my umbrella is only marginally worse than that, so I'll assign it a utility of +9. The worst outcome is if it rains and I don't take my umbrella; I'll assign it a utility of -10. The outcome in which it rains and I do take my umbrella is somewhere in the middle, so I'll assign it a utility of 0.
These utilities can be expressed in a table:
The numbers provide an indication of my feelings towards the various possible outcomes. Some readers may be skeptical that precise figures can be assigned to subjective feelings like this. However, the precise values of these figures are not important; all that is important is their relative values. It is clear from these figures that in fine weather I prefer not to have to carry an umbrella, but only slightly, and that I have a fairly strong preference for fine weather over rainy weather. I could equally well have used other numbers to express these preferences. It is usual to use positive numbers for good outcomes and negative numbers for bad outcomes, but this is not necessary.
So should I take my umbrella? Well, it depends on how likely it is that it will rain today. Suppose the weather forecast tells me that there is a 20% chance of rain today. Then I can calculate the expected utility of taking my umbrella; it is (0 x 1/5) + (9 x 4/5) = 7.2. Similarly, the expected utility of not taking my umbrella is (-10 x 1/5) + (10 x 4/5) = 6. Since the expected utility of taking my umbrella is greater, I should take my umbrella. Again, notice that the absolute value of these numbers is unimportant; only their relative value is significant.
Strictly speaking, the conclusion that I should take my umbrella only follows under the assumption that I want to maximize my expected utility. Some people take this as a descriptive principle about humanity; it is a true description of human beings that they want to maximize their expected utility. Others take it as a normative principle about rationality; if you are rational, you should attempt to maximize your expected utility. Still others subscribe to neither of these principles.
Still, whether or not either of these principles hold, it is uncontroversial that there are circumstances in which people want to maximize their expected utility--for example, when I am deciding whether or not to take my umbrella. I may not explicitly go through the above calculation, but something like it goes through my mind. If the chance of rain is high enough, I take my umbrella because I don't want to get wet, and otherwise I leave it at home because I can't be bothered carrying it.
Nobody needs an expected utility calculation to tell them whether or not to take an umbrella, but they can be help in situations where more is at stake. For example, suppose you are considering whether to have your child vaccinated against whooping cough. The whooping cough vaccination protects children from a potentially fatal disease, and also has some rare but serious side-effects. Let us suppose that unvaccinated children have a 1 in 50,000 chance of dying of whooping cough, which is reduced to 1 in 1,000,000 by vaccination. Also, suppose that the vaccination carries a 1 in 200,000 chance of causing permanent brain damage. (If you were to carry out this calculation seriously, you would first need to find some reliable estimates for these probabilities; my figures are not reliable!)
To carry out the calculation, you need to assign utilities to the possible outcomes. Suppose you assign a utility of -10 to your child's death, -8 to permanent brain damage, and 0 to normal health. (Remember that the absolute size of these numbers is arbitrary; only their relative size matters. A utility of -10 in this example clearly doesn't mean the same as a utility of -10 in the previous example!) Then the expected utility of vaccinating your child is (-10 x 1/1,000,000) + (-8 x 1/200,000) = -1/20,000. The expected utility of not vaccinating your child is -10 x 1/50,000 = -1/5,000. Since the expected utility of not vaccinating your child is lower (a larger negative number), then to maximize your expected utility you should vaccinate your child.
Fred is short-sighted, and he is considering laser surgery to correct his vision. He does some research, and finds that the chance of a complete correction is 46%, the chance of a partial correction is 44%, the chance of no change in vision is 9% and the chance of a worsening of vision of 1%. Fred assigns a utility of +5 to achieving a complete vision correction, +2 to a partial correction, 0 to no change in vision, and -10 to a worsening of vision. He also assigns a utility of -2 to the cost and physical discomfort of the operation. Assuming Fred wants to maximize his expected utility, should he have the operation?
There are many pitfalls in reasoning about risks. See if you can tell which of the following arguments are correct, and which are fallacies.