Uncertainty: In simplest terms, uncertainty refers to degree to which something is unknown or not understood. Uncertainty in risk assessment is measured by Bayesian probability, defined as the degree of belief that a proposition is true (e.g., the tree is alive). Probability is expressed on a scale of 0 to 1. Zero probability indicates certainly false, one indicates certainly true, and 0.5 indicates that true and false are equally likely.

Uncertainty arises from two major sources:

• The first source is variation. For example, there are 52 unique cards in a regular deck of playing cards, representing four suits and 13 ranks. The variation among cards determines the probabilistic outcomes of many card games, e.g., the probability of drawing an Ace at random is 4/52 = 7.7%. The same is true of natural systems; for example, the probability of choosing a beetle-infested tree at random from a forest depends on the variation among trees and the level of beetle infestation in the forest. Variation can often be measured, which leads to empirically derived measures of probability.
  
• The second source of uncertainty is ignorance. Ignorance is a more subjective form of uncertainty in that it depends primarily on perspective. For example, if I stand next to a tree and can examine it directly, I will assign a different probability to to the event, "beetle infested," than if I look at the same tree in an aerial photograph.

Variation and ignorance can often be mixed in real-world problems; calculations of probability make no distinction between the two. The distinction is important, however, because they are sensitive to different factors and modes of investigation.

Value: Value reflects the worth, usefulness, or importance of a particular object or service. Goods and services that are traded as commodities can be assigned monetary values. For many ecological components, monetary values are not easily defined and it is useful to describe value in terms of utility. Utility is simply an arbitrary measurement scale that can be used to compare objects. Two outcomes having the same utility are said to be equivalent in terms of value, and a decision maker would have no preference for one over the other.

Risk: Risk is often used synonymously for probability when adverse outcomes are discussed, e.g., "the risk of being struck by lightning is small." In CRAFT, we use the more formal definition of risk as expected loss. It is calculated as the integrated sum across all possible outcomes of the probability of an outcome times its value. For example, if there are two possible outcomes with equal probabilities, the expected loss or risk is simply the average value of the two outcomes. For multiple possible outcomes with different probabilities, risk is the weighted average value, where probabilities are used as weights.

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