Probability- Basics

 Probability

The likelihood of an event occurring 

What is an Event

For Understanding, We take the example of flipping a coin

There will be 2 Chance 

Head  <-----------------> Tails

Chance occurring is an event

So, the event will be likely or unlikely to happen

For Event Occurring --> 1

For Event not Occurring ---> 0

Define Formula

A --> Event'

P(A) --> Probability

For Coin  

               P(A) = 1/2= 0.5

For Dice 

             P(A) = 1/6 =~ 0.167

So, Now We have to see the Probability of Two independent events same time

            P( A and B ) = P(A). P(B)

Expected Values

The average Outcome we expect if we run an experiment many times

There are two kinds of Probability

1. Experimental Probability - 80 % 

2. Theoretical ( True) Probability- 20%

Experimental Probability – The probability we assign an event, based on an experiment we conduct

They have approximate Values

Trial – Observing an event occurs and recording the outcome

Experiment – A collection of one or multiple trials.

Examples

In Cards

A--> ♠ ( Spade) 

E(A) = Outcome we expect in an experiment

E(A) = P(A) x n

n = 20

P(A) = 13/52= 0.25

0.25 x 20 = 5 

We Expect Spade 5 times in total 20 Chances

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In Archery

A= 10          P(A)= 0.5                  

B= 20          P(B)= 0.4

C= 100        P(C)= 0.1

E(X)=?

E(X)= A. P(A) + B.P(B) + C.P(C)

= 0.5 x 10 + 0.4 x 20 + 0.1 x 100

= 5 + 8 + 10 =  23 ~20 

We will never get to this point in a circle

So What does it mean

Probability is about the chance of happening 

In today's world, we need predictions

So, We can use expected values to make predictions about the future based on past data

Weather Predictions use these techniques-- As they always use the terms " Up to " and " Expected "

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Now Rolling 2 Dice and Sum the Outcome

P(7) = 6/36= 1/6  -- seems less chance?

So What about other's sum probability 

 Probability frequency distribution

A collection of the probabilities for each possible outcome of an event

Frequency--> Frequency is the number of times a given value or outcome appears in the sample space.

Frequency Distribution Table--> It is a table matching each distinct outcome in the sample space to its associated frequency

                                      

                                     

Why do we need frequency distributions?

We need the probability frequency distribution to try and predict future events when the expected value is unattainable. 

Expected Value

E(X)= P(2).2 + P(3).3 + P(4).4 + P(5).5 + P(6).6 + P(7).7 + P(8).8 + P(9).9 + P(10).10 + P(11).11 +  P(12).12

= 1/36 x 2 + 1/18 x 3 + 1/12 x 4 + 1/9 x 5 + 5/36 x 6 + 1/6 x 7 + 5/36 x 8 + 1/9 x 9 + 1/12 x 10 + 1/18 x 11 + 1/36 x 12

= 1/18+1/6+1/3+5/9+5/6+7/6+10/9+1+6/5+11/18+1/3

=~ 7.3 ~ 7 

Complement

Simply Everything the event is not

A'

A + A' = Sample Space

P(A) + P(A’) = 1

(A’)’ = A

Example:

Assume event A represents drawing a spade, so P(A) = 0.25. 

Then, A’ represents not drawing a spade, so drawing a club, a diamond or a heart. 

P(A’) = 1 – P(A), so  P(A’) = 0.75

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Let's Understand the probability of not getting a 3 on Dice

P(A) = P(1)+P(2)+P(4)+P(5)+P(6)

P(A)= 

P(3)= 1 / 6

1-1/6= 5/6