16.5: Doing Better by Using Frequencies

A good deal of research has shown that we reason more accurately about many probabilities, including the probabilities of conjunctions, if we think in terms of frequencies or proportions or percentages, rather than simply in terms of probabilities. Recall Linda, the single, outspoken, bright, philosophy major. When people are asked whether it is more probable (or more likely) that she is (1) a bank teller, or (2) a bank teller who is active in the feminist movement, well over half of them usually (incorrectly) select (2).

But when people approach the same problem in terms of percentages or frequencies, they do better. If we keep the same profile, but rephrase the two questions to ask: what proportion or percentage of a group of one hundred randomly selected women who fit this profile are (1) bank tellers, and what proportion or percentage of a group of one hundred randomly selected women who fit this profile are bank tellers who are active in the feminist movement, more people avoid the conjunction fallacy. More people (correctly) select (1)—although there is still a strong tendency to commit the conjunction fallacy.

This tendency also shows up in frequency versions of the conjunction fallacy. The question, “Are there more six letter words ending in ‘ing’ than having ‘n’ as their fifth letter?” is a question about relative frequencies. Many still say that there are more ‘ing’ words, even though every six letter word ending in ‘ing’ has an ‘n’ as its fifth letter, and there are also non-‘ing’ words with ‘n’ in fifth place (e.g., ‘barons’). Still, many of us do better here if we think in terms of percentages, proportions, or frequencies than if we simply think in terms of probabilities, or if we simply think in terms of probabilities.

In short, one of the best ways to improve your accuracy in estimating probabilities is to rephrase things in terms of frequencies whenever you can. Instead of asking how probable it is that a person with a given set of symptoms has a disease, ask, “what proportion of people in a randomly selected group of 100 who have these symptoms have this disease?” What is the frequency of this disease in a group of 100 people who have these symptoms. In fact, you don’t even need words like ‘frequency’ or ‘percentage.’ Just ask: about how many people out of a hundred (or a thousand) who have these symptoms also have the disease. You can then translate your answer into percentages or probabilities very easily.

It may also help to use percentages instead of probability numbers. Rather than saying the probability of having the disease is .2, you can say that the probability is 20%. When you ask how many things out of a hundred have a certain property and use percentages, then the percentages translate directly into number of things; 90% is just 90 of the items out of 100.

Thinking in terms of percentages or frequencies also makes it easier to think about cumulative risk. If the probability of a certain brand of condom failing is 0.01, ask: how many times out of 100, or 1000, would it fail. The respective answers are 1 every 100 times, and 10 every thousand times. So, over the longer run, there is a substantial chance of failure.