How to calculate probability?

In our day to day life we use many words about the chances  of occurrence of events.

Probability theory attempts to quantify these chances of occurrence or non occurrence of events.

Probability : Numbers Related problems

Problems based on number probability are explained here with the help of different examples. 

We learn here how to make different digit numbers from given numbers.

Short symbols for

1. Number of probability = n(p)

2. Expected number of Events = ENE

3. Total Number of Events = TNE

n(p) = ^ENE/ _TNE

For Example

How many new numbers can be formed using 4,5,7,4,8,6,3,5,5?

Case 1:

To calculate  probability when numbers can take any position

Sol:

Total digits(4,5,7,4,8,6,3,5,5) = 9

Occurrence of number 4 =2

Occurrence of number 5 =3

Expected number of Events (ENE)

                         = ^9!/_2!.3!

                       
                         =^{9×8×7×6×5×4×3!}/_2×1×3!

                                                   
                       =30240

New numbers which can be formed using 4,5,7,4,8,6,3,5,5(when numbers can take any position) =30240

Case 2:

To calculate probability when all Even Numbers lie together

Sol:

Number are=5,7,3,5,5,4486(here all even numbers counted as 1)

Total digits(5,7,3,5,5,4486) = 6

Occurrence of number 4 = 2

Occurrence of number 5 = 3

Expected number of Events = ENE 

                                           =^6!/_3! × ^4!/_2!

                                  =^6×5×4×3!/_3! × ^4×3×2!/_2!               
                            =120× 20

                                           =1440

Case 3:

To calculate probability when No Even Numbers lie together

Sol:

Numbers which can be formed using 4,5,7,4,8,6,3,5,5 (when numbers can take any position) - Numbers which can be formed using 4,5,7,4,8,6,3,5,5(All Even Numbers lie together)

          =30240-1440

          =28800

Case 4:

To calculate probability when all Odd Numbers lie together

Sol:

Number are  = 57355,4,4,8,6 (here all odd numbers counted as 1)

Total digits(57355,4,4,8,6) = 5

Occurrence of number 4 =2

Occurrence of number 5 =3

Expected number of Events = ENE= ^5!/_3! × ^5!/_2!

                                            ₌^5×4×3!/_3! × ^5×4×3×2!/_2! 

                                                       =20× 60

                                                        =1200

Case 5:

To calculate probability when No Odd Numbers lie together

Numbers which can be formed using 4,5,7,4,8,6,3,5,5 (when numbers can take any position) - 

Numbers which can be formed using 4,5,7,4,8,6,3,5,5(All Odd Numbers lie together)

                                                                                 =30240-1200

                                                                                 =29040