Probablity Theory — Counting — I

Probability Theory — Definition

Generally when a survey is made, it is done for a small samle instead of full population. Recent relevant example is Covid-19 vaccine. It needs to be given for a entire population but cant test the safety of the vaccine against entire population. Hence a small sample of population is selected and efficiency is tested.

Now we need to count what probability of this sample experiment is going to be effective for entire population . Hence counting is very much needed , now how do we calculated counting.

Say if u have a dice of 6 sides and want to know on what chance does the dice output has. It can be either 1 || 2 || 3 || 4 || 5 || 6.

The counting was easy rite

What if u need to pick 4 cards from a deck of 52 cards and how many chances the 4 cards will be Ace . Its impossible to calculate all possible combinations mentally. we need some calculating methodology or principles to do it.

We can understand counting by starting simple. what is the number between 1 and 100. It is easy to calculate total is 100. If you need a rule for it.

Rule 1: The number of numbers between 1 and N is N

What if the number doesnt start from 1

Say the problem is find the numbers between 12 to 100. we all know it is

100–11 = 89

So how do we phrase a formula for it

n = 100 (Ending number)

k = 12 (Starting number)

Total No’s = n-k+1

which is

100 -12 + 1 = 89

Rule 2 : The number of numbers between n and k is (n- k + 1)

Lets make it more complicated

what if we need numbers between 12 to 100 which are divisible by 9

We need only certain numbers between 12 and 100 .

technically they are 18,27…99

so we can apply rule 2 if they are consecutive numbers i.e., 1,2,3 e.t.c. How can we change 18,27,…99 to 1,2,3

By dividing them with 9

it becomes 2,3,…11

Now applying rule 2 of n-k +1

which is 11–2+1 = 10

Lets make it more && more complicated

the numbers are in the sequence of 14,17,20…95,98

the are not in sequence, not divisible by any number… But they have a common difference . Increase by 3. So I can add or remove it by a number to make it divisible by 3.

i can add 1 so that it becomes

15,18,21…96,99

and now divide it by 3

5,6,7..32,33

n = 33

k = 5

Applying Rule 2 = n-k + 1=33–5+1=29