A risk is formally defined by combining the likelihood of an event with the magnitude of its outcomes, which makes risk a quantitative object rather than a vague feeling of danger. Because the definition requires specifying both probability and consequence, disagreements about risk often trace back to which outcomes were counted and over what time horizon.
Choosing what counts as a harm, whose exposure matters, and which time frame to use are value judgments made before any measurement begins, so no risk estimate is purely technical. Two honest analysts can produce different numbers simply because they bounded the problem differently.
To choose among risky options one can multiply each outcome's value by its probability and sum, selecting the option with the highest expected value. This rule treats a small chance of a large loss as commensurable with a large chance of a small loss, which can clash with how people actually weigh catastrophic but rare events.
Lay judgments of how risky something is are shaped by features beyond the numbers, such as whether exposure is voluntary, controllable, familiar, dreaded, or potentially catastrophic. A hazard that kills few people can be judged far riskier than a statistically deadlier one because it rates high on dread and involuntariness.
Effective risk communication starts by identifying which few facts would actually change a recipient's decision, then tests whether messages convey those facts as intended rather than assuming comprehension. Flooding people with accurate but irrelevant information can leave them less able to act than a focused message would.
Experts and the public often disagree about risks not because the public is simply irrational but because each attends to different aspects: experts to annual fatality counts, laypeople to features like catastrophic potential and control. Treating the gap as mere public ignorance misdiagnoses it and leads to communication that fails.