Probability

In general, probability is a numerical representation of how likely is the occurrence of certain observations.

Whenever an empirical investigation involves uncertainty, due to sampling, insufficient understanding of the actual procedure or of the laws governing the observed phenomena, or for any other reason, the concept of probability may be applied.

The concept of probability developed from the investigation of the properties of various games, like rolling dice, but in addition to the very practical desire of understanding how to win, it also incorporates deep philosophical thought. Currently, most scholars consider probabilities as objectively existing values that can be best revealed by observing long sequences of potential occurrences of events and using the relative frequencies of the events to approximate their probabilities. Another, also objective, view of probability is that it can be calculated as the ratio of the possible number of observations when an event occurs to the total number of possible observations. Some other scholars think that probability is subjective: It expresses the degree of belief a certain person has in the occurrence of an event. Fortunately, all these approaches lead to probabilities that have essentially the same properties.

A simple illustration is rolling a fair die with all outcomes being equally likely. Then, each outcome has probability equal to 1/6. More precisely, this may be a subjective belief of an observer, or the assumption of the experimenter, or may be empirically tested by observing the results of long sequences of rolls. In either case, the probability of having a value less than 4 (i.e. 1, 2, or 3) is equal to Vi, and the probability of having an even outcome (i.e. 2, 4, or 6) is also Vi. The probability of having an outcome less than 4 and even is 1/6, as this only happens for 2. There are however, events that are possible but cannot occur at the same time. For example, having a value less than 4 and greater than 5 is not possible. The event that cannot occur is called the “impossible event” and is denoted by Ø.

Probability theory has many applications in the social sciences. It is the basis of random sampling, where the units of the population of interest, usually people, are selected into the sample with probabilities specified in advance. The simplest case is simple random sampling, where every person has the same chance of being selected into the sample and the steps of the selection process are independent from each other. In such cases, probability models with all outcomes having equal probability may be relevant, but the more complex sampling schemes often used in practice require other models. Probabilistic methods are also used to model the effects of errors of a measurement and, more generally, to model the effects of not measured or unknown factors.