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Sample space of a random experiment is the collection or set of all possible outcomes of that experiment. It is written in the form of a set denoted as S, wherein all possible outcomes are listed as the set elements.

Event is the subset of all possible outcomes of the experiment.

Equally likely events are two or more events that have an equal chance of occurrence

Independent events are events that occur without being influenced or affected by the occurrence of any other event.

Mutually exclusive events are two or more events that cannot occur at the same time

Probability Probability quantifies the chance of a particular event occurring. Expressed as a numerical value between 0 and 1, where 0 represents impossibility (Impossible event) and 1 signifies certainty (Sure event), which provides a measure of uncertainty regarding the outcome of a random experiment or uncertain situation. The probability of an event A is denoted by P(A) and is calculated as the ratio of the number of favorable outcomes n(A) to the total number of possible outcomes n in the sample space. \[P(A)=\frac{\mathrm{n}(A)}{\mathrm{n}}\] The fundamental principles of probability theory govern the analysis of random phenomena, guiding the understanding of uncertainty in diverse fields such as statistics, mathematics, science, economics, and everyday decision-making.

Probability: Complement determines to the probability of the non-occurrence or opposite of a specific event. If the probability of event A is denoted as P(A), then the complement of A, often represented as A′, is the probability that A does not occur P(A′). \[P(A^{\prime})=1-P(A)\]

P(A ∩ B) is the probability of both independent events A and B happening together. It is calculated as the product of the individual probabilities P(A) and P(B). \[P(A \cap B)=P(A) \cdot P(B)\]

P(A ∪ B) is the probability of either event A or event B (or both) occurring. \[P(A \cup B)=P(A) + P(B) - P(A \cap B)\] Note that, for mutually exculsive events P(A ∩ B) = 0 thus \[P(A \cup B)=P(A) + P(B)\]

Conditional probability is a measure of the likelihood of one event happening, given that another event has already occurred. It is denoted as P(A/B) representing the probability of event A occurring under the condition that event B has taken place, where the two events are dependent. \[P(A| B)=\frac{P(A \cap B)}{P(B)}\]

Note: For any two independent events the following conditions must be satisfied
P(A ∩ B)=P(A).P(B) or P(A)=P(A/B)