A probability is a number from 0 (impossible) to 1 (certain), often written as a percentage. A fair coin has a probability of 0.5, or 50 percent, of landing heads. A few simple rules go a long way.

Independent events

Two events are independent if one does not affect the other, like two tosses of a coin. To find the chance that both happen, multiply. The chance of two heads in a row is one half times one half, so the chance that two fair coins both land heads is one in four.

For the chance that something happens at least once, it is often easiest to find the chance that it never happens and subtract from 1. The chance of rolling no six on one die is 5/6. In four rolls, it is 5/6 multiplied by itself four times, which is 625/1,296, or about 0.48. So the chance of at least one six in four rolls is about 1 minus 0.48, or about 52 percent.

Coincidences are more common than intuition suggests. In a group of just 23 people, the chance that at least two share a birthday is slightly better than even, because there are 253 different pairs of people, each a chance for a match.

Conditional probability

A conditional probability is the chance of something given that something else is true. The key warning: the probability of A given B is usually not the same as the probability of B given A. Most professional basketball players are tall, but most tall people are not professional basketball players. Confusing the two directions is behind many errors in medicine and law.

False positives in medical tests

Imagine a screening test for a disease that affects 1 percent of the people tested. The test catches 90 percent of people who have the disease, and it wrongly gives a positive result to 5 percent of people who do not. You test positive. How worried should you be?

Count 10,000 people. About 100 have the disease, and the test catches 90 of them. The other 9,900 are healthy, and 5 percent of them, or 495, also test positive. That makes 90 + 495 = 585 positive results, of which only 90 are true. So only about 15 percent of positive results are true (90 divided by 585 is about 0.154).

The test is not useless; a positive result raised the chance from 1 percent to about 15 percent. But the healthy group is so much larger that even a small error rate produces many false alarms. This is why positive screening results are usually followed by a second, more specific test.

Regression to the mean

Any result that depends partly on luck tends to be followed by one closer to average. This is regression to the mean: extreme results tend to be followed by more ordinary ones. In the 1880s Francis Galton noticed that very tall parents tended to have children who were tall, but on average less extremely tall than themselves.

Regression creates false lessons. Kahneman told of flight instructors who believed that praise made trainee pilots worse and criticism made them better. An excellent landing was usually followed by a less excellent one, and a bad landing by a better one, whatever the instructor said. Excellent flights were often followed by more ordinary ones anyway, praise or not, because each landing involved some luck.

The same trap appears whenever we act after an extreme result. A city puts speed cameras at the intersections with the most crashes last year; next year crashes there fall. Some of that fall may be regression, since a bad year is often followed by a more normal one. Careful evaluations compare against similar places without the change.

Worked example: what a second test does

Return to the screening example from this level: 10,000 people tested, 1 percent with the disease, a test that catches 90 percent of cases and gives false positives to 5 percent of healthy people.

First test. 90 true positives and 495 false positives, for 585 positive results. The chance that a person with a positive result has the disease is 90 out of 585, about 15 percent.

Second test. Now suppose all 585 people take a second, independent test with the same accuracy.

• Of the 90 who have the disease, 90 percent test positive again: 81 people.
• Of the 495 healthy people, 5 percent test positive again: about 25 people (24.75).

About 81 + 25 = 106 people test positive twice. Of these, 81 have the disease, so the chance is 81 out of 106, roughly 76 percent, or about three in four.

What changed? The second test was applied to a group in which the disease was far more common, about 15 percent rather than 1 percent. The same test is much more informative when the starting probability is higher. That is why doctors consider symptoms and risk factors when interpreting a result.

Two cautions. First, this assumes the two tests make errors independently. If the same quirk that fooled the first test also fools the second, repeat testing helps less, which is why follow-up tests usually use a different method. Second, ten of the 100 sick people were missed by the first test and are not in this group at all.

The key habit is to count people, not percentages. Natural frequencies turn a confusing problem into simple addition and division.