Probability Terminology
The following are Probability terminology that are helpful in understanding the concepts of probability and rules of probability for solving different probability-related real-life problems.
Sets: A set is a well-defined collection of distinct objects. The objects making up a set are called its elements. A set is usually capital letters i.e. $A, B, C$, while its elements are denoted by small letters i.e. $a, b, c$, etc.
Null Set: A set that contains no element is called a null set or simply an empty set. It is denoted by { } or $\varnothing$.
Subset: If every element of a set $A$ is also an element of a set $B$, then $A$ is said to be a subset of $B$ and it is denoted by $A \ne B$.
Proper Subset: If $A$ is a subset of $B$, and $B$ contains at least one element that is not an element of $A$, then $A$ is said to be a proper subset of $B$ and is denoted by; $A \subset B$.
Finite and Infinite Sets: A set is finite, if it contains a specific number of elements, i.e. while counting the members of the sets, the counting process comes to an end otherwise the set is infinite.
Universal Set: A set consisting of all the elements of the sets under consideration is called the universal set. It is denoted by $\cup$.
Disjoint Set: Two sets $A$ and $B$ are said to be disjoint sets if they have no elements in common i.e. if $A \cup B = \varnothing$, then $A$ and $B$ are said to be disjoint sets.
Overlapping Sets: Two sets $A$ and $B$ are said to be overlapping sets, if they have at least one element in common, i.e. if $A \cap B \ne \varnothing$ and none of them is the subset of the other set then $A$ and $B$ are overlapping sets.
Union of Sets: The Union of two sets $A$ and $B$ is a set that contains the elements either belonging to $A$ or $B$ or both. It is denoted by $A \cap B$ and read as $A$ union $B$.
Intersection of Sets: The intersection of two sets $A$ and $B$ is a set that contains the elements belonging to both $A$ and $B$. It is denoted by $A \cup B$ and read as $A$ intersection $B$.
Difference of Sets: The difference between a set $A$ and a set $B$ is the set that contains the elements of the set $A$ that are not contained in $b$. The difference between sets $A$ and $B$ is denoted by $a-b$.
Complement of a Set: Complement of a set $a$ denoted by $\bar{A}$ or $A^c$ and is defined as $\bar{A}=\cup$.
Experiment: Any activity where we observe something or measure something. An activity that results in or produces an event is called an experiment.
Random Experiment: An experiment, if repeated under identical conditions may not give the same outcome, i.e. The outcome of a random experiment is uncertain, so that a given outcome is just one sample of many possible outcomes. For the random experiment, we know about all possible outcomes. A random experiment has the following properties;
- The experiment can be repeated any number of times.
- A random trial consists of at least two outcomes.
Sample Space: The set of all possible outcomes in a random experiment is called sample space. In the coin toss experiment, the sample space is $S=\{Head, Tail\}$, in the card-drawing experiment the sample space has 52 members. Similarly the sample space for a die={1,2,3,4,5,6}.
Event: Event is simply a subset of sample space. In a sample space, there can be two or more events consisting of sample points. For coin, the list of all possible events is 4, found by $event=2^n$, that is i) $A_1 = \{H\}$, ii) $A_2=\{T\}$, iii) $A_3\{H, T\}$, and iv) $A_4=\varnothing$ are possible event for coin toss experiment.
Simple Event: If an event consists of one sample point, then it is called a simple event. For example, when two coins are tossed, the event {TT} is simple.
Compound Event: If an event consists of more than one sample point, it is called a compound event. For example, when two dice are rolled, an event B, the sum of two faces is 4 i.e. $B=\{(1,3), (2,3), 3,1)\}$ is a compound event.
Independent Events: Two events $A$ and $B$ are said to be independent if the occurrence of one does not affect the occurrence of the other. For example, in tossing two coins, the occurrence of a head on one coin does not affect in any way the occurrence of a head or tail on the other coin.
Dependent Events: Two events A and B are said to be dependent if the occurrence of one event affects the occurrence of the other event.
Mutually Exclusive Events: Two events $A$ and $B$ are said to be mutually exclusive if they cannot occur at the same time i.e. $A\cup B AUB=\varnothing$. For example, when a coin is tossed, we get either a head or a tail, but not both. That is why they have no common point there, so these two events (head and tail) are mutually exclusive. Similarly, when a die is thrown, the possible outcomes 1, 2, 3, 4, 5, 6 are mutually exclusive.
Equally Likely or Non-Mutually Exclusive Events: Two events $A$ and $B$ are said to be equally likely events when one event is as likely to occur as the other. OR If the experiment is continued a large number of times all the events have the chance of occurring an equal number of times. Mathematically, $A\cup B \ne\varnothing$. For example, when a coin is tossed, the head is as likely to occur as the tail or vice versa.
Exhaustive Events: When a sample space $S$ is partitioned into some mutually exclusive events, such that their union is the sample space itself, the event is called an exhaustive event. OR
Events are said to be collectively exhaustive when the union of mutually exclusive events is the entire sample space $S$.
Let a die is rolled, the sample space is $S=\{1,2,3,4,5,6\}$.
Let $A=\{1, 2\}, B=\{3, 4, 5\}$, and C=\{6\}$.
$A, B$, and $C$ are mutually exclusive events and their union $(A\cup B \cup C = S)$ is the sample space, so the events $A, B$, and $C$ are exhaustive.