Introduction of set theory and probability:

Set Theory: A collection of well defined objects is called a set.

For example,

  • The set of all natural numbers,
  • The set of all equilateral triangles in a plane,
  • The set of all real numbers,
  • The set of all vowels in English alphabet are some of the examples of sets since we can undoubtedly say what objects are there in each of the collections. Study of set is known as set theory.

Probability: The chance for happening is known as probability.

For example,

  • our cricket team is likely to win the world cup,
  • The prices of essential commodities are likely to be stable are some of the examples of probability.

 

Examples on Set theory:

 

Example 1:

Consider two sets A = {2, 1, 5, 8, 9} and B = {2.6,-1, -2.4}. Prove A U B = B U A.

Solution:

A U B   = {2, 1, 5, 8, 9} U {2.6,-1,-2.4}

= {2, 1, 5, 8, 9, 2.6,-1,-2.4} ------->1

B U A   = {2.6,-1,-2.4} U {2, 1, 5, 8, 9}

= {2.6,-1,-2.4, 2, 1, 5, 8, 9} ------->2

From 1 and 2, it is prove that A U B = B U A.

Example 2:

Consider two sets A = {2,-1, 4,-3,-7,-1.2} and B = {2, 5,-7, 3.6}. Prove A`nn`B = B`nn`A.

Solution:

            A`nn`B  = {2,-1, 4,-3,-7,-1.2}  {2, 5,-7, 3.6}

                      = {2,-7} ----------> 1

            B`nn`A  = {2, 5,-7, 3.6} {2,-1, 4,-3,-7,-1.2}

                      = {2,-7} ---------->2

From 1 and 2 it is proved that A`nn`B = B`nn`A.

 

Examples on Probability:

 

Example 1:

 Two coins are tossed simultaneously. What is the probability of getting two heads?

Solution:

In tossing two coins the sample space S = {HH, HT, TH, TT}, n(S) = 4. Let A denote the event of getting two heads A = {HH}, n (A) = 1.

Therefore n (A) = P (A) / n(S) = 1/ 4.

 

Example 2:

An integer is chosen at random from 1 to 50. Find the probability that the number is divisible by 5.

Solution:

Sample space S = {1, 2, 3… 50}, n(S) = 50.

              Let A denote the event of getting a number divisible by 5.

              So, A = {5, 10, 15, 20, 25, 30, 35, 40, 45, 50}, n (A) = 10.

              n (A) = P (A) / n(S) =10/ 50 = 1 / 5.

These are the examples of set theory and probability.