Humans will never have a complete understanding of everything that happens on the Earth’s surface, and so it is often convenient to resort to thinking in terms of probabilities. In principle one could completely characterize the physics of a human hand and a coin, but in practice it is much more productive to assign probabilities to the outcomes of a coin toss. In similar fashion spatial analysts may avoid the virtual impossibility of predicting exactly where landslides will occur by assigning them probabilities based on patterns of known causes, such as clay soils, rainfall, and earthquakes. A map of probabilities assigns each location a value between 0 and 1, forming a probability field.
Such a map considers only the probability of a single, isolated event, however. The probability that two points a short distance apart will both be subject to landslide is not simply the product of the two probabilities, as it would be if the two outcomes were independent, a conclusion that can be seen as another manifestation of Tobler’s First Law. For example, if the probability of a landslide at Point A is ½ and at Point B a short distance away is also ½, the probability that both will be affected is more than ½x½=¼, and possibly even as much as ½. Technically, the marginal probabilities of isolated events may not be as useful as the joint probabilities of related events — and joint probabilities are properties of pairs of points and thus impossible to display in map form unless the number of points is very small.