In real life, an event is something that happens. In philosophy, events are timeless, unique, self-contained entities in time or timeless instantiations of human qualities in objects. On some theories, only such changes in the shape of getting or losing a physical property to the effect of other properties can constitute events, such as the drying of the earth’s soil. Some philosophers, however, believe that the changes that have occurred to the earth’s surface over recent centuries can be considered events, and that there is certainly a continuity of events going on behind the scenes. For this school of thought, all changes, including those that have occurred due to man’s activities, are events.

Let’s take one example that is easy to visualize and describe. Let’s say that the earth’s orbit is tilted slightly off its axis, and it moves in a certain way around the sun, causing it to undergo an elliptical orbit. Now suppose that a butterfly is flying through the air, and that it stops on a nearby flower. According to this theory, this is an elementary event, and that no matter how many times this occurs, the probability of the particular event itself occurring in any particular instance is always the same.

The problem comes up when we consider the many complex events that can occur. If the earth’s orbit is shifted by only a degree, then the probability of its happening again in any given instance, according to this kind of thinking, is simply the chance density, which is the probability density of each single instance, multiplied by the number of times the earth shifts its orbit. It therefore follows that there must be a binomial tree with infinitely many degrees of sequence (a so called infinite trees) from which we can extract our real root mean, whatever it may be, and we must be able to say something about the real value of this mean. And this, if we follow the philosophy of probability, is equivalent to saying that any real event can be expressed numerically, by means of a finite set of outcomes.

Probability theory is a tool, used in mathematics for the analysis of frequencies and differences in quantities and frequencies of other variables. For instance, the frequency with which a bell chimed in unison with another bell, or the frequency with which an angry mob erupted, are subsets of the event mentioned earlier; although they are not actually themselves subsets of the event itself. The probability density formula says that these subsets are independent events, and hence their probability values will never be the actual values of their own frequencies, but rather the values of their intersections. So, for example, if we plot out the probability of the two bells chiming at different frequencies, then we have to graph them separately on their own subsets of the interval between their frequencies, as if they were themselves a frequency with which they were coinciding. We would see that the first bell chimed, on its own subsets of the interval, had a higher probability than the second bell.

The probability density function is a mathematical tool which enables us to express probabilities in mathematical notation. A normal integral formula tells us how many times something occurs but never expresses the likelihood of how often something occurs. In the language of mathematics, we may say that the expected value of an event, once called an occurrence, is the sum of all possible values of an event at every point in the sample space. We may also say that it’s the set of all possible events times the slope of the probability curve over time. The probability density function expresses the probability of an event occurring at a particular value over a period of time.

A probabilistic analysis is a method of predicting (with reasonable accuracy) the behavior of a complex system under specified parameters. It may be used to evaluate a system’s response to external perturbations, such as changing input data, before a set up is achieved; or for experimental studies, such as testing reactions of interest under various conditions. This type of experiment is sometimes referred to as a “conditional random sampling” because, though the observations make sure events occur regularly, the measurements help identify the kind of events which are likely to occur.