Probability theory is a branch of mathematics that studies the likelihood of random phenomena. A random event’s outcome cannot be predicted in advance, but it may be one of many potential outcomes. The final result is thought to be decided by chance.
In everyday speech, the term “probability” has a variety of meanings. Two of these are especially critical for the development and application of probability theory. One understanding is that probabilities are relative frequencies, as observed in basic games ranging from coins and dice, to complex games like poker and blackjack. The distinguishing characteristic of chance games is that the outcome of any given trial cannot be predicted with confidence, even though the aggregate outcomes of a large number of trials exhibit some predictability. For example, the assertion that the likelihood of “heads” equals one-half when flipping a coin implies that the relative frequency with which “heads” actually occurs would be approximately one-half over a large number of tosses, although it contains no inference about the outcome of any given toss.
There are several related examples involving groups of individuals, gas molecules, and genes. Actuarial estimates of a person’s life expectancy at a given age represent the cumulative knowledge of a large number of people but do not purport to predict what will happen to any single individual. Similarly, estimates about the likelihood of a genetic disorder occurring in a child of established genetic makeup are claims about the relative frequency of incidence in a large number of cases, and not about a particular person.
Fundamental Rules of Probability
There are four fundamental laws of probability. Typically, these laws are expressed more rigorously than required for the purposes of this paper, using set theory and probability notation.
- The First Rule: The first rule states that all probabilities are, by convention, integers between 0 and 1. A probability of 0 indicates an improbability, while a probability of 1 indicates a certain probability. Most interesting events have odds in between these two extremes.
- The Second Rule: This rule states that events are said to be “disjoint” if they do not share any outcomes. For instance, the event of a patient developing cancer is distinct from the event of the same patient not developing cancer, since the two cannot occur concurrently. In the other hand, the event of cancer is not dissociable from the event of cancer with metastases, since all cases include the presence of cancer. If two events are disjoint, the probability that one or both of them occurs is equal to the number of their individual probabilities. For example, if the likelihood of rain is 0.5 (meaning 50%) and the probability of thunderstorm is 0.3, then the probability of either rain or thunderstorm must be 0.8, or 80%.
- The Third Rule: If one may list all possible disjoint events in an experiment, the likelihood of at least one of them occurring is one. For instance, if a patient is diagnosed on a 5-point scale with 1 indicating no disease, 2 indicating probable disease, 3 indicating uncertain disease status, 4 indicating probable disease, and 5 indicating definitely disease, the likelihood that one of these states is chosen is 1.
- The Fourth Rule: This states that if two events are separate (i.e., knowing the outcome of one does not provide information about the likelihood of the other occurring), then the probability of both occurring is provided by the product of their individual probabilities. Thus, if the likelihood that findings on an MR picture will result in a diagnosis of a malignant tumour is 0.1 and the probability that it will rain today is 0.3 (a seemingly unrelated occurrence to the MR imaging results), the probability of both a malignant tumour and rain today is 0.1*0.3 = 0.03, or we can say 3%.
In summary, probability distributions for events often adhere to these four benign rules, which can most likely be deduced with theoretical knowledge. Such likelihood estimates may be beneficial in clinical settings, for example, when determining the probability of a particular diagnosis based on the results of one or more diagnostic tests. Numerous probability equations used in clinical research are conditional probabilistic in nature.