Risk is a concept that everyone understands and almost no one evaluates very well. Risk is a function of both the likelihood that something will happen and the consequences that result if it does. When evaluating risks we always seem to be more focused on one or the other.
For example, driving without your seatbelt entails a risk. Those who do usually focus on the likelihood of getting into an accident: “I’m just going around the corner”, rather than the consequence, which could be serious injury or death. Conversely, the decision to buy a lottery ticket is usually based on the consequence, “I could be a millionaire!”, rather than the likelihood of it happening, which is extremely low.
Correctly evaluating risk requires an honest consideration of both likelihood and consequence. Statisticians, mathematicians and actuaries examine joint probabilities to evaluate risk. Simply put, they multiply the probabilities together to evaluate the possible outcomes. So if you are faced with a financial investment choice that has a 25% chance of paying $10,000 or a 10% chance of paying $25,000, your financial risk is the same (because 0.25 x 10,000 and 0.10 x 2,500 both equal 2500, your theoretical pay-off). Simple.
Things get complicated because there is often uncertainty (or in a mathematical sense, distributions) associated with both the likelihoods and consequences, and complex decisions often involve a cascade of nested or otherwise related risk assessments.
Keeping the math straight is the job of decision models, and making it transparent is the job of (good) modelers.