What is Probability?

Generally, probability is the study of the chances of events occurring. In mathematics, probability is the measure of the chance that particular event(s) will occur, expressed on a linear scale from 0 (impossible) to 1 (certain). Probabilities can only be defined for events. Events in mathematics are a little different from how we typically think of them, and different schools of thought in probability have differing opinions of how to interpret events.

There are two major interpretations for probability theory, that of the frequentists and Bayesians. Frequentists see probabilities as describing tendencies in repeated trials. Both schools agree that probabilities are a good predictor of long-running frequencies, but Bayesians hold that it does not always make sense to think of probability in terms of repeated trials if we are interested in events that can only occur once.

• Frequency View

The probability of an event $A$ is the expected or estimated relative frequency of $A$ in a large number of trails. The proportion of the number of event $A$ occurs in n trials is expected to be roughly equal to the theoretical probability $P(A)$ if n is large. $P(A)$ is a single number, and describes the relative frequency by which event A will occur as the number of trials approaches infinity. There is only one, true, single $P(A)$, which does not change depending on the information given, and which good analysis should seek to find.

$$P_n(A) → P(A) for large n$$

For example, if we are rolling one dice for 6000 times, the dice shows $4$ happens $1000$ times out of $6000$ trails. Thus, the relative probability of the event that the dice shows $4$ is $\frac{1000}{6000} = \frac{1}{6}$ from the perspective of objectivists. According to the objectivists, we can be certain that the true probability of rolling a $4$ actually is $\frac{1}{6}$, and is actually found via these repeated trials.

• Bayesian View

The bayesian view, sometimes called the opinion view, arose in response to the many cases where relative frequency does not meet certain demands. Gamblers, for example, need to make judgements without first experimenting on the relative frequency of certain outcomes. In this view, statements of fact about probability ultimately reduce to some kind of initiative judgement of the uncertainties involved. Such judgements are also be called probabilistic opinion or degrees of belief since they mostly rely on subjective opinion and only contain superficial objective quality. There isn’t necessarily a ‘true’ $P(A)$, just a certain amount of confidence that $A$ will occur. Bayesian theory is concerned with making sound judgements under circumstances where repeated trials are impossible.

For example, if you are the patient considering an operation, you want the doctor to tell you what he thinks your chances are. Since the notion of repeated operations make no sense and the state of health varies from person to person, it is difficult for the doctor to know survival percentages for you. Also different doctors might have different opinions and then somehow you form your own opinion as to your chances. The probabilistic opinion in this case is often necessary although imprecise, and requires reading into causal circumstances, weighing them by confidence, and ultimately making a justified judgement, rather than a statement of fact.