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Introduction to Probability

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Probability is an important branch of Mathematics. Let us talk about introduction to Probability. Probability includes the quantitative analysis of large Sets of data.
Some definitions related to the Probability Theory are given below-
· An experiment is a situation that involves chances or probability to occur that leads to the outcomes.
· An outcome refers to the result of a single trial of an experiment.
· The term event refers to the one or more outcomes of the same experiment.
· A Sample Space contains the Set of all possible outcomes of an experiment.
Above are the points, which give us brief probability introduction.
Probability of an event ‘A’ can be defined mathematically as-
P(A) = The number of ways that event ‘A’ can occur / The total number of outcomes that are possible.
The probability of an event ‘A’ is the Ratio of the number of ways in which event ‘A’ can occur to the total number of possible outcomes of the experiment. It is the numerical measure of the likelihood of the occurrence of an event.
The value of the probability for occurrence of an event always lies between 0 and 1. It can't be more than one.
0 ≤ P(A) ≤ 1.
The sum of all Probabilities in the sample space is equals to 1.
The probability of any event that cannot be in the sample space is zero.
The probability of an event that is not occurring can be calculated by subtracting it from 1.
P(A') = 1 – P(A).
A probability space refers to the uncertainty of the experiment. Sample space is the set of outcomes while probability measure is the real function of the subset of sample space.
Suppose if the experiment is to deploy a smoke detector and see if it works then there could be four outcomes, so the sample space is-
S = (fire, smoke), (no fire, smoke), (fire, no smoke), (no fire, no smoke).
These are the mutually exclusive outcomes. This is all about intro to probability.

Sample Space

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Sample space Probability is basically a Set of all possible outcomes of any experiment. It is denoted by S and sometimes called as an event too. Each outcome individually is called as element. It can be containing many Numbers of outcomes.
If Outcomes number is finite, we call it as finite sample space
Some examples of Finite Sample Spaces are:
· Rolling a Dice
· Tossing a Coin
· Draw a Card
When Outcome is not discrete and Continuous then those types of sample spaces are called as Indefinite Sample space
Examples of Indefinite Sample Spaces:
· Human Height
· Length of a Chord
· Flight Arrival Time

Now understand probability sample space with the help of examples:

Example1: What will be the sample space when we will toss a Coin?
Solution: When we will toss a coin there are two probable outcomes head (H) and tail(T)
So sample space (S) = (Head, Tail)

Example 2: What will be the sample space when we will toss a Coin?
Solution: When we will toss two coins there are two probable outcomes For Coin 1: head (H1) and tail(T1)
Coin2: head (H2) and tail(T2)
Sample Space(S) = (H1, H2),(H1, T2), (T1, H2)(T1,T2)
So suppose sample set for n numbers of coins will be 2n

Example3:There are 6 Numbers on a Dice which are 1, 2 , 3 , 4 , 5 , 6 .What will be the Sample space for a single throw ?
Solution: Sample space (1 , 2 ,3 ,4 , 5 ,6)

Example4: What will be the Sample Space to pull a sock from a drawer containing 15 red socks, 10 green socks, and 6 yellow socks?
Solution” S=Red, Green , yellow
Example5: What is the probability of getting an even number when rolling a die?
P = P(2) + P(4) + P(6)

Example 6: What will be the Sample Space to pick a ball from a drawer containing 15 yellow balls, 10 white balls, and 16 red balls?
Solution” S= Yellow, white, red