Americans, as any ScienceBlogger will tell you, have a woefully poor understanding of math and science. For the most part, even the most ignorant among us are able to stumble through life, but what happens when we’re confronted with a genuine scientific question with a real impact on our lives?

Consider the typical doctor’s office scenario: the doctor asks a breast cancer patient to decide on a treatment. “There’s a 30 percent chance of recurrence in five years,” she tells the patient, “but with chemotherapy, the chance is reduced to 10 percent.” If the patient doesn’t have a basic understanding of probability, she can’t make an informed decision about whether to undergo treatment.

Doctors are likely to be better educated than the vast majority of their patients, so how does this discrepancy impact the way they share information about medical risk with their patients? Andrea Gurmankin Levy and Jonathan Baron devised a study to try to understand the difference between doctors’ and patients’ concept of medical risk.

They asked paid participants to rate seven different medical conditions on a scale of “badness” from 0 to 127, where 100 corresponds to a 100 percent chance of immediate death (the scale goes up to 127 to account for the possibility that some conditions may be worse than death). The conditions were: a harmless, painless wart on the big toe; amputation of a big toe; deafness in one ear; amputation of one leg below the knee; amputation of both legs above the knee; blindness; and blindness and deafness. Nearly all participants ranked these conditions in the same order I’ve just listed them.

Next, the participants rated the badness of a series of probabilities of developing each condition on the same scale. For example, in one question they were asked to rate the badness of a 10 percent chance of having a toe amputated. Here are the results:

They are as you might expect: as the probability of developing the condition increases, the badness response also increases. The worse the condition, the higher the badness response at any level of probability.

But how do these ratings compare to the ratings of doctors? Perhaps surprisingly, the results weren’t especially different. Doctors tended to be more sensitive to probability, rating the higher probabilities as worse and lower probabilities as better than laypeople — but this effect was primarily observed at older ages. Young laypeople weren’t significantly different from doctors.

What’s most striking is the individual differences between respondents. Take a look at this chart:

The numbers have been transformed to a new scale in this chart, but you can still read it the same way. The dotted lines represent the bottom 20 percent of respondents in terms of sensitivity to probability, while the solid lines represent the top 20 percent. The bottom fifth of respondents essentially made no distinction between a 0.1 percent chance of acquiring a condition and a 100 percent chance — all were seen as equally bad. Meanwhile, the top fifth saw a 32 percent chance of deafness in one ear as worse than a 3.2 percent chance of death.

It’s pretty clear that the bottom twenty percent of this group simply has no grasp of probability, but the top twenty percent may be making a similar mistake by weighting the probabilities too much. The problem for doctors then becomes, how to express probabilities in a form that all their patients can understand. Unfortunately, this study doesn’t provide clear answers to that question.

Gurmankin Levy, A., & Baron, J. (2005). How bad is a 10% chance of losing a toe? Judgments of probabilistic conditions by doctors and laypeople.

Memory & Cognition, 33(8), 1399-1406.