In real life situations many events happen. Some events happen for sure, under any circumstance. For example sun rises in the morning. Similarly you are sure that it is impossible for some events to happen like you cannot roll on number 7 when a fair die is thrown. But the occurrences of most of the events in everyday life happen with certain amount of uncertainty. The amount of certainty is defined as the probability for the event to occur. In other words, the probability is a measurable quantity and hence mathematical concept is involved. The major tool that is used from the concept of mathematics is the set theory. In this article, let us study how these two theories are related.  

Probability Definition

Probability is defined has the number of favorable out come of the total out come.

Probability Formulas

P (Any Event)

1) P(x) = $\frac{Favorable\ out\ come}{Total\ out\ come}$

2) P(A/B) = $\frac{P(A∩B)}{P(B)}$

3) P(B/A) = $\frac{P(A∩B)}{P(A)}$

4) P(A∩B) = P(A) x P(B)

Probability Examples

Below you could see probability examples

Solved Examples

Question 1: Find the probability of getting 1 red ball out of 5 green balls and 6 red balls.
Given: There are 5 green balls and 6 red balls

P(red balls) = $\frac{6C_{1}}{11}$

= $\frac{6}{11}$

Question 2: Find the probability of getting 1 red marble and 1 green marble from a set of 7 red marbles and 3 green marbles.
Given: There are 7 red marbles and 3 green marbles.

P(1 red and 1 green marble) = P(A) x P(B) = $\frac{7C_{1}}{10}$ x $\frac{3C_{1}}{10}$ 

= $\frac{7}{10}$ x $\frac{3}{10}$

= $\frac{21}{100}$

Probability Theory

We have briefly explained what a probability is. To give a mathematical definition in a simple way, a probability is defined as the number of favorable outcomes out of a total of all possible outcomes. A coin is tossed and it is a legendary question whether it is going to fall heads up or tails up. You are not sure of any of these two outcomes. But certainly one out of the two outcomes is bound to happen. This leads to the probability of the desired event (say your favorite is ‘head’) to occur is ½. At the same time it also implies a connection to the set theory.

Set theory deals with all possible operations of different sets. Let us first define what a set is. A set is nothing but a collection of items. If your desired item is one of the elements in that set, then the item is called a ‘sub set’ of the main set. Now, from this simple example one can clearly realize the connection  between the probability concept and the basic concept of sets. Even when the probability functions become complicated, the relation between the two concepts exists strongly. The combined approach to both these two concepts can be termed as probability set theory.

Probability Notation

In any mathematical operation, notations or symbols play a very important role. The various concepts in probability are easily expressed in set notations. Hence it is imperative to know the different notations that are used in set theory. Let us study about the various notations right from the fundamentals.

As mentioned earlier, a set is a data in the form of collection of items. The number of items is not relevant. The number 1, a single item can be called as a set and at the same time all number from 1 to infinity can also be called a ‘set of natural numbers’. The items are described between the curly brackets { }.
  • A set is normally denoted by a capital letter. For example, letters like A, B, C etc. can be used to denote a set.

  • A set can also be empty. That is the set does not contain any element. An empty set is denoted by the Greek letter $\phi$.

  • A set which contains all the elements that is considered for the required purpose is called an universal set and is denoted by U. In other words, the set that contains all possible outcomes is a universal set. For example, the universal set for studying the probability of any even with rolling a die is, U = {1, 2, … 6}. The following table will elaborate the different set notations.

Probability Set Notation

x $\epsilon$ A                          
Element ‘x’ belongs to set A
A U B  Union of set A and set B.
That is, the elements of set A and elements of set B. Common elements on both  sets are considered as singles.
A ∩ B   Intersection of set A and set B. That is, the elements which are common to both the sets A and B.
A ∩ B = $\phi$  The sets A and B are disjoint. That is, there are no  common elements between sets A and B
A'  Complement of set A. That is, the elements which do not belong to set A but are part of the universal set U.

Probability Sets

Now, let us study some set notations that are used in probability operations.

Suppose A represents the set of desired outcomes and S represents the total outcomes (sample space), then the event E is denoted as A subset of S. But in any sample space, there may be different events. For example, in rolling a standard die, the different events may be,

1)    E(N) = Rolling on a particular number ‘n’. Then the set N is {n} where 1≤ n ≤ 6}  
2)    E(P) = Rolling on a prime number. Then the set P is {2, 3, 5}
3)    E(E) = Rolling on an even number. Then the set E is {2, 4, 6}
4)    E(O) = Rolling on an odd number. Then the set O is {1, 3, 5}

In all the above cases the set of the sample space S is {1, 2, 3, 4, 5, 6} and all the sets N, P, E and O are subsets of set S. The different probabilities are denoted as the ratios of the numbers in relevant subset to the numbers in the set A. For example, the probability of getting a prime number is,

P(P) = [Total number of elements in subset P / Total number of elements in set S] = $\frac{3}{6}$ = $\frac{1}{2}$

In the same way many probability relations are denoted in set notations. They are better termed as probability set functions. We will discuss about them in detail in the next section.

Probability Set Function

Now In the previous section we have explained the fundamental function of a probability set. Let us now try to study some more important functions.

The probability of a certain event in a sample space S is always 1. That is P(A) = 1. In probability set this is expressed functionally as,
A is a subset of and equal to set S

Similarly, the probability of an impossible event is always 0. That is, P(A) = 0. That is expressed as,

A ∩ S = $\phi$

If only one of the events A and B can occur at the same time, then the sets of A and B are disjoint. Then,

P(A) ∩P(B) = 0

The union of two probabilities A and B is given by the function,

P(A U B) = P(A) + P(B) – P(A) ∩P(B)

As a corollary, the union of two exclusive events A and B is,

P(A U B) = P(A) + P(B)

Sometimes, a probability of event A is determined when the probability of a related event B is given. It is called the conditional probability of A given the probability of B already occurred. It is denoted as P(A/B) and the functionally expressed as,

P (A/B) = [P(A ∩ B)/P(B)] or P(A ∩ B) = P(A/B)*P(B) provided event B is not an impossible event. (That is, P(B) $\neq$ 0)

Two events A and B are called independent events if, P(A/B) = P (A).

Therefore, in case of independent events A and B, P(A ∩ B) = P(A)*P(B)