What psychologists call apophenia — the human tendency to see connections and patterns that are not really there — gives rise to conspiracy theories. It is also at work, though usually in a milder form, in our perceptions about cancer and our revulsion to randomness.

It takes several mutations, in specific combinations, for a cell to erupt into a malignant tumor. The idea that random copying errors are prominent among them is thoroughly mainstream. What was new about the paper was its attempt to measure this biological bad luck and see how it compares with the two other corners of the cancer triangle: environment and heredity — mutations we inherit from our parents that can give cancer a head start.

The mix of these influences varies. A lifetime of heavy smoking has been shown to multiply the risk of lung cancer — the most common malignancy in the world — by some twentyfold, or about 2,000 percent. But that is an anomaly. One of the great frustrations of cancer prevention has been the failure to find other chemical carcinogens so definitive or damaging, especially in the dilute amounts in which they reach most of the public.

For a handful of cancers, biological agents are important, like human papilloma virus in cervical cancer and helicobacter pylori in stomach cancer. On another level, inflammation and hormonal imbalances, like those associated with obesity and diabetes, can drive cells to multiply more frequently, increasing the chance of mutations causal and accidental.

Finally, heredity — like the BRCA mutations involved in some breast cancers — can have a profound effect in individual cases. But inheritance appears to be involved in just 5 to 10 percent of all cancers.