[Maths Class Notes] on Probability and Statistics Pdf for Exam

In our daily life, we are often confronted by our own ignorance. Sometimes we ponder about tomorrow’s weather or upcoming elections but we fail to get the outcome with certainty. And then we do the guessing game. Probability is a precise mathematical rule that helps us to understand and analyze our ignorance. It may not answer the questions of tomorrow’s weather or the results of an upcoming election but provide us with a framework that can work with our limited knowledge and make sensible decisions based on what we fail to know. Probability teaches us the ideas of randomness, prediction, expected value, and estimation in a more sensible and mathematical way.

Probability Meaning in Statistics

Both Probability And Statistics are the two most important concepts in Mathematics. On one hand, Probability is all about chance, and on the other hand, Statistics is about how we handle various data with the help of different techniques. It helps to simplify extremely complicated data in a very easy and understandable way.

Whenever we are doubtful about the outcome of an event, we talk about the probabilities of certain outcomes. The analysis of such events that are governed by probability is called statistics.

If someone tells you the probability of something happening, so basically they are telling you how likely that something is to happen. When people buy lottery tickets, the probability of their winning is usually stated, and sometimes, it might turn out something like 1/10,000,000 (or even worse). This tells you that it is not very likely that you might win.

Probability Formula 

The formula for probability can tell us how many choices we have over the total number of possible combinations. Therefore, the formula can be written as:

[{text{probability}} = frac{text{possible choices}}{text{total number of options}}]

Statistics is a science that is concerned with studying methods and its development for collecting, analyzing, interpreting, and presenting empirical data.

What is Statistics?

Statistics is a science that is concerned with studying methods and its development for collecting, analyzing, interpreting, and presenting empirical data.

The two fundamental ideas in the field of statistics are uncertainty and variation.

Any measurement or data collection effort which is subjected to a number of sources of variation and gets repeated, then the answer would most likely change. Therefore Statisticians attempt to first understand and then control the sources of variation in any given situation.

What is Statistical Inference? 

Statistical inference deals with making statements or inferences about the characteristics of the true underlying probability measures. It is very obvious that these inferences are based on some kind of information that the statistical model is a part of it. Another important part of the information is given by an observed outcome or response, which is usually referred to as the data. 

Statistical Formula

There are a few statistical formulas to help us solve statistical problems. The formulas are listed below:

Consider x as the item provided and n as the total number of items. 

Mean : [frac{text{sum of all the terms}}{text{total number of terms}}]

Median: M = [(frac{n+1}{2})^{th}]      if n is odd.

M = [frac{(frac{n+1}{2})^{th} term + (frac{n}{2} + 1)^{th} term}{2}]     if n is even.

Mode: Most frequently occurring value.

Terms Used in Probability and Statistics

Here are various terms that are utilized in the probability and statistics concepts. :

1. Random Experiment: An experiment whose result we cannot predict until and unless it is noticed is known as a random experiment. The most basic example can be throwing a dice randomly. It is certain that the result will be uncertain to us. The output can be any number between 1 to 6. Therefore, this experiment is a random experiment.

2. Sample Space: A sample space is a set of all the possible results or outcomes of a random experiment. It is a set of all the possible outcomes that we can get when we throw a dice. 

3. Random Variables: These are the variables that denote the possible outcomes of a random experiment. They can be of two types: Discrete Random Variables and Continuous Random Variables. Discrete random variables can take only distinct values which are countable. Continuous random variables can take an infinite number of possible values.

4. Independent Event: When the probability of occurrence does not impact the probability of another event, then both the events are called independent events. For example, when you flip a coin along with throwing a dice, the probability of getting a ‘head’ will be independent of the probability of getting a 6 in dice.

5. Mean: The mean of a random variable is an average of the random values of the possible outcomes of a random experiment. 

6. Expected Value: The expected value is the mean of a random variable. For example, when we roll a dice having six faces, the expected value will be the average value of all the possible outcomes, i.e. 3.5.

Solved Examples

Example 1: If there are 5 marbles in a bag out of which 4 are blue and only 1 is red. What is the probability of blue marbles being picked up?

Solution 1: The number of blue marbles is 4 and the total number of marbles are 5. Therefore, by using the formula: 

[{text{Probability}} = frac{text{possible choices}}{text{total number of options}}]

= [frac{4}{5}] = 0.8

Example 2: Find the mean of 8, 11, 6, 22, 3.

Solution 2) Using the formula of mean:

Mean = [frac{text{sum of all the numbers}}{text{total numbers}}]

= [frac{8 + 11 + 6 + 22 + 3}{5}]

= 10.