As a number of psychologists have shown, people tend to understand frequencies better than probabilities. Thus "event A occurs in 3 outcomes out of 10" is better understood than "the probability of the event A is 30 percent." Even frequencies are misunderstood by some.
Told that cancer X kills 2,850 of 10,000 people and that cancer Y kills 28.5 out of 100 people, many believe Cancer X to be worse than Y. Presumably even worse would be a cancer that killed 28,500 out of 100,000. This tendency to be impressed by the larger numbers is sometimes called ratio bias.
Probabilities and frequencies, even if well-understood, can give rise to counterintuitive results. This is especially relevant to the issue of false positives in mammography, but I'll state it generally.
Let's consider yet another cancer Z and a test for it that satisfies the following three conditions:
1.) The probability a person has cancer Z is 1 percent.
2.) If the person has Z, the test is positive 95 percent of the time.
3.) If the person doesn't, the test is still positive 3 percent of the time.
Presented as frequencies the conditions are:
1.) On average 1 out of 100 people have Z.
2.) Of 100 people with Z, 95 will test positive.
3.) Of 100 people who are Z-free, 3 of them will test positive.
However these conditions are presented, the crucial question is what fraction of those people who test positive for Z actually have it. The surprising answer (see below) is about 24 percent, a calculation that studies show many doctors are unable to perform.
Statistical terms from p-values to odds ratios and confidence intervals are often misinterpreted by patients and doctors alike. (This gives an unfortunate new meaning to a double-blind test.) Probability and statistics are often seen as cold subjects, not mindful of the individual.
Uncertainty and trade-offs, however, are an inevitable part of life, and a proper and humane understanding of them can help minimize the number of patients turning prematurely cold.
Answer: I'll assume readers do not suffer from ratio bias and that tests for Z are given to 100,000 people. By assumption, 1 percent of them or 1,000 (.01 x 100,000) will have Z. Since 95 percent of these 1,000 will test positive for Z, there will be approximately 950 (.95 x 1,000) positive tests. But 99,000 (100,000 - 1,000) people are healthy.
Nevertheless, by assumption 3 percent of them or 2,970 (.03 x 99,000) will also test positive. These latter will be false positives.
Thus, we have a total of 3,920 positive tests (2,970 + 950) of which only 950 are true positives. That is, only 950 of the 3,920 positive tests indicate cancer.
In other words, the probability of cancer Z given that one has tested positive for it is only 950/3,920 or 24 percent.
John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers, "Innumeracy" and "A Mathematician Reads the Newspaper," as well as (just out in paperback) "Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.