A probability experiment refers to the analysis that depicts the possibility an event occurring in the future through the performance of a series of examination. Based on the results of a probability experiment, one can ascertain the truthfulness of some results or the chances of a certain event of interest occurring.

The determination of a result in a certain probability experiment is termed as a trial. Some of the examples of probability experiments could be the act of throwing a dice or revolving a spinner.

In probability theory, an event refers to the set of results of an experiment that can take probability values. This means that an event is a subset of the sample space that takes probabilistic values. Considering a finite sample space, an event would be any subset of the sample space. Similarly, the subset elements of the sample space are events since they belong to the universal set.

In instances where the sample space is infinite, the outcomes would be values that lie within the real number line and thus the approach of the probabilistic event will fail. There exist two types of events. These are the simple and complementary events. The simple events, also known as elementary events are events that have a single outcome in the sample space.

The probabilities of simple events are always positive, zero or undefined. On the other hand, the complementary events are events whose occurrence of one event hinders the occurrence of the other event. Similarly, events could be regarded as independent or dependent events.

In this regard, the occurrence of independent events does not affect the occurrence of the other event. Concerning the dependent events, the occurrence of one event is subject to the occurrence of another event.

Probability refers to the chance or possibility that an uncertain event will occur. Therefore, probability estimates the likelihood of the occurrence of events under conditions of uncertainty. A number between one and zero depicts probability. At times, a percentage value may be used to indicate a probability. A zero probability for an event implies that the event will not happen.

A probability of one for a particular event implies that its occurrence is definite. When calculating probability, the ratio between the numbers of occurrence of an event with the total number of set of outcomes has to be obtained. Additionally, in determining the probability value, one could adopt either theoretical or experimental probability.

Theoretical probability entails the calculation of probability based on the number of occurrence of an event dived by the total outcomes. An example of theoretical probability would involve rolling a six-sided die on the side indicated 3.

The sample event would be the 3 while the outcomes would be values between 1 and 6. Thus, the probability of occurrence of 3 would be 1 out of 6 implying it is a sixth. On the contrary, experimental probability refers to the number of times of occurrence of an event divided by the number of trials undertaken.

A sample space denoted by Ω or *S* refers to the combination every possible outcome expected in a probability experiment. An example of a sample space in tossing a coin twice would be where H denotes the head while T denotes the tail. A sample space subset indicates an event in the countable samples.

The process of defining the events from a sample event entails constituting of an σ-algebra to outline clearly the events. Despite the fact that σ-algebra is the general form of describing events within a sample space, it is usually theoretical since it defines the subsets of interest in relevance.

For the probability theory to be standardized, there exist properties, which must be satisfied. These properties relate to the nature of probability experiments, events, sample space, and probability values. Initially, the probability of events needs to satisfy the following properties given the denotation of the probability of φ by P (φ).

First, the range of probability should be 0≤P(φ)≤1 Secondly, the probability of the sample space, Ω, is know with surety as P (Ω) =1. Thirdly, the sigma-additive should hold.

This will imply that for two events, A and B, their union would be φ denoted as A∩B=φ which would imply P(AUB) =P(A) +P(B). Lastly, the finite additive principle states that the probability of occurrence of disjoint events is equal to the sum of the probability of the two events.

Similar to the above properties, the other sub-section properties include the probability of an empty set, complement events, union, and the monotonicity of probability. The probability of an empty set is zero.

The probability of complement events portrays that the sum probability of two complementary event is equal to one that is P(A) =1-P(A^{c}). The probability of a union of two events A and B is equal to sum of the probability of A and B less the intersection of A and B.

This can be denoted as P(AUB)=P(A)+P(B)-P(A∩B). This property replicates to any number of events. The property of the monotonicity of probability imply that for two events, A and B, where one is the subset of the other, then P(A)≤P(B). This property can be demonstrated by the range property where P (A∩B^{c})≥0. Thus P(A) =P(B) +P(A∩B^{c}) ≥P(B).

Two events are independent if the occurrence of one event does not affect the occurrence of the other event. This implies that the following conditions are essential for events to be independent. Given two events, A and B, the probability of occurrence of event A should be equal to probability of A given occurrence of B. This can be illustrated as P(A/B) =P(A).

Secondly, the probability of the occurrence of event B should be equal to the probability of B considering the occurrence of event A. Finally, the probability of A union B should be equal to the product of the probabilities of the two events.

Therefore, P(AUB)=P(A)*P(B). For example, in an experiment involving the throwing a dice, the appearance of a three-sided face in the first trial does not affect the occurrence of any face in the second trial.

During the sampling process, with or without replacement processes determine the independence of events. Sampling with replacement involves that the possibility of one element of the group being chosen more than once. This indicates that during the process, events are termed as independent since the occurrence of one outcome does not affect the probability of the second occurrence during the next trial.

Sampling process without replacement entails the effects of the occurrence of the first outcome on the possibility of the occurrence of the next outcome in the subsequent trial. Thus, such a sampling process involves dependent events. In situations where it is difficult to determine the nature of the events, it is essential to regard them as dependent until the otherwise established.