Essentials of Data Science – Probability and Statistical Inference - Set Theory and Events

In the previous note on Essentials of Data Science – Probability and Statistical Inference – Sample Space and Events, we have seen how to define and set up experiments, sample space, and how events are associated with sample space. This note will further extend the concept of sample space and events and learn how to utilize set theory concepts such as union, intersection, and complement to find the relationship or the association among the events.

Probability is very important concept and it can be interpreted in a number of ways. One of the ways it can be interpreted using operations of set theory.

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1 Set Theory & Probability Theory

1.1 Union of Events

1.2 Intersection of Events

1.3 Difference of Events

1.4 Complement of Event

1.5 Subset of Events

1.6 Disjoint Events

1.7 Mutually Disjoint Events

2 References

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Set Theory & Probability Theory

A set is a collection of objects, and these objects are called the elements of the set. If S is a set and x is an element of S, we say x $\in$ S. Even it is possible that a set does not contain any element. Those type of set is called an empty set.

Union of Events

Suppose A and B are any two events of a sample space $\Omega$ . Let us consider a new event C = A $\cup$ B. It is a union of the events A and B. The union of A and B consist of all the outcomes that either in A or in B or in both A and B.

Union of two events — A union of two events

Example 1:

If the outcome of an experiment consists in the determination of the gender of a newly born child, then $\Omega$ = {M,F} where M and F indicate Male and Female child, respectively.

If A = {M}, then A is the event that the child is a male (boy). Similarly, if B = {F}, then B is the event that the child is a female (girl). Thus A $\cup$ B = {M, F}, i.e., A $\cup$ B is the whole sample sapce $\Omega$ . This is also called as sure event.

Intersection of Events

Suppose A and B are any two events of a sample space $\Omega$ . Let us consider a new event C = A $\cap$ B. It is an intersection of events A and B. The intersection of A and B consist of all the outcomes that are in both A and B. It means, event A $\cap$ B will occur if both A and B occur.

Intersection of two events — The intersection of two events

The intersection of events A $\cap$ B is the set of all simple events of A and B which occurs when the simple events of A and B occur.

Difference of Events

Suppose A and B are any two events of a sample space $\Omega$ . Let us consider a new event C = A – B. It means, the event A – B contains all simple events of A, which are not contained in B. This event can also be read as A but not B or A minus B occurs, if A occurs but B does not occur. Also A – B = A $\cup B^c$

It is an intersection of events A and B. The intersection of A and B consist of all the outcomes that are in both A and B. It means, event A $\cap$ B will occur if both A and B occur.

Difference of two events — A different of two event

Complement of Event

Suppose A is an event of a sample space $\Omega$ and $A^c$ is another event which contains all the simple events or outcomes which are not the part of event A. The complementary event of A whenever A does not occur.

Subset of Events

It is also possible that an event A can be subset of another event B. It is represented as A $\subseteq$ . It means, all the simple events of A are also part of the sample space of B.

Disjoint Events

Two events A and B are disjoint if A $\cap$ B = $\emptyset$ holds, i.e. if both events cannot occur simultaneously.

Mutually Disjoint Events

The events $A_1, A_2, A_3, ... ,A_m$ are said to be mutually or pairwise disjoint, if $A_i \cap A_j = \emptyset$ whenever i $\neq$ j = 1,2,3,….,m.

Example 1:

Rolling a die: If a die is rolled once, then the possible outcomes are the numbers of dots on the upper surface: 1,2,3,4,5,6.

Sample space is the set of simple events and these are $\omega_1$ = “1”, $\omega_2$ = “2”, $\omega_3$ = “3”, $\omega_4$ = “4”, $\omega_5$ = “5”, $\omega_6$ = “6”. Thus $\Omega$ = { $\omega_1, \omega_2, \omega_3,\omega_4,\omega_5,\omega_6$ }.

Suppose if A = { $\omega_1, \omega_2, \omega_3, \omega_4, \omega_5$ } and B is the set of all odd numbers, then B = { $\omega_1, \omega_3, \omega_5$ } and thus B $\subseteq$ A.

Suppose if A = { $\omega_2, \omega_4, \omega_6$ } is the set of even numbres and B = { $\omega_3, \omega_6$ } is the set of all numbers which are divisible by 3, then A $\cup$ B = { $\omega_2, \omega_3, \omega_4, \omega_6$ } is the collection of simple events for which the number is either even or divisible by 3 or both.

Suppose if A = { $\omega_1, \omega_3, \omega_5$ } is the set of odd numbers and B = { $\omega_3, \omega_6$ } is the set of the numbers which are divisible by 3, then A $\cap$ = { $\omega_3$ } is the set of simple events in which the numbers are odd and divisible by 3.

Suppose if A = { $\omega_1, \omega_3, \omega_5$ } is the set of odd numbers and B = { $\omega_3, \omega_6$ } is the set of the numbers which are divisible by 3, then A – B = { $\omega_1, \omega_5$ } is the set of simple events in which the numbers are odd but not divisible by 3.

Suppose if A = { $\omega_2, \omega_4, \omega_6$ } is the set of even numbers, then $A^c$ = { $\omega_1, \omega_3, \omega_5$ } is the set of odd numbers. Then the events A and B are disjoint.

References

Essentials of Data Science With R Software – 1: Probability and Statistical Inference, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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Essentials of Data Science – Probability and Statistical Inference – Set Theory and Events