Statistics and Probability

What is probability?

Probability meaning - It is the chance of an event happening. The concept of probability is used to predict the likeliness of an event. The probability of an event lies between 0 and 1, and the higher the probability, the more likely that event will happen. For instance, the probability of getting Head in flipping a coin is ½ or 50 %. The probability that the value of a random variable will be less than or equal to a particular outcome is given by the probability distribution function.

What is Statistics?

Statistics meaning - It is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data.

Types Of Statistics

Inferential statistics, which employs descriptive statistics to test hypotheses and draw conclusions, and descriptive statistics, which describe the characteristics of sample and population data, are the two main branches of statistics.

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Why Probability and Statistics?

In daily life, we always try to make the best decisions in which we think will succeed. Probability and Statistics give you one way to think about the decisions you make and give you one direction to think about your decision. Using Statistics to record and analyse data, finding out probabilities for different situations or cases, and then trying to make out the best rational decision possible, this method is used by all scientists, engineers, companies, and gamblers.

Today statistics and Probability is used in all sports in making strategies, also analysing the performance of sportspeople. For example, Average, Strike rate in Cricket and Average Goals per match in football and baseball, and all the numerical data you see in the newspaper.

Have you ever wondered,‘How Google Maps works?’

Google Maps gives you real-time traffic updates. It also tells you the shortest distance and time that you will take to reach your destination.

Google Maps collects your data, like the time you take to go through a place, the distance you travel, and your path between two locations, through many ways like Satellite, GPS, Location Services on your phone, and Google Map Makers. At the same time, Google is collecting huge amounts of data from other people around you or going through the path you want to go. Then google analyses all these data statistically and gives you the most probable time, distance with different routes, and mode of transportation you will take between the two locations.

Sometimes Google Maps is wrong, you have to consider the fact that Earth is big. With the Collection of more and more data, Statistical Analysis becomes better and better, and so does google maps.

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Probability and Statistics For Career

If you are preparing for entrance exams like SATs, JEE(main, advanced) or preparing for the GMAT, GRE, probability statistics cover an important portion of the mathematics syllabus. Also, there is a lot of career scope in Probability and Statistics as all companies hire data analysts to work efficiently and to improve.

Some Basic Concepts

The basic formula of probability Theoretical Probability =(Number of Favourable Cases/Total number of possible cases)

For Example

1. Getting a probability of getting 4 or 6 in rolling a 6 faced dice is ? as the number of favourable cases is 2, i.e., {4,6} and total number of possible cases are 6, i.e., {1,2,3,4,5,6}.

2. My friend Paul wanted to bet with me whether Messi will score a goal in the next match or not. He had 8 goals in 10 matches. So Paul said he surely will. I am not a big Messi fan and I did not want to be biassed so I used Statistics and Probability. Although he had 8 goals, he scored that in 3 matches(3,2,3) and no goals in the remaining 7 matches. The theoretical probability I get is 0.3. So I decided to bet against it. What would you do if you were in my situation?

Also read| NCERT Exemplar Class 9 Maths Solutions Chapter 14 Statistics and Probability

Independent Events:

Two or more events are said to be independent if the occurrence or non-occurrence of any of them does not affect the probability of occurrence or non-occurrence of other events. In other words, two or more events are said to be independent if the occurrence or non-occurrence of any of them does not influence the occurrence or non-occurrence of other events.

For example, when a coin is tossed twice, the event of the occurrence of the head in the first throw and the event of the occurrence of the head in the second throw are independent events.

Dependent Events

Suppose we have 10 green balls and 10 orange balls in a box. We pull out(draw) one ball, which may be green or orange. Now there are 19 balls left in the box. What is the probability that the second ball will be orange?

It depends, If the first ball was orange, then the bag is left with 9 orange balls out of 19 so the probability of drawing an orange ball on the second draw is 9/19. But if the first marble we draw is green, then there are still 10 orange balls in the bag and the probability of drawing an orange ball out of the bag is 10/19.

A second draw is a dependent event as we can see it depends upon the first draw.

Conditional Probability

If A and B are two events associated with the same sample space of a random experiment, then the conditional probability of the event A under the condition that the event B has occurred. In other words, event B has already happened, now what is the chance of event A?

Conditional Probability is written as P (A | B), and it is given by

Let us understand this by taking one example when a die is thrown, it's sample space S = {1, 2, 3, 4, 5, 6}. Let event A be an occurrence of a number greater than 4 is {5, 6}. The event of the occurrence of an even number is B = {2, 4, 6}. Then P(A|B) is the probability of occurrence of a number greater than 4 when an even number has occurred i.e. one of 2, 4, 6 has already occurred. out of these three numbers 2, 4, 6, only 6 is greater than 4. Hence, here when an even number has occurred, the total number of cases is only 3 (not 6) and the favourable number of cases is only one because out of 2, 4, 6 only 6 is greater than 4.

Also, read| NCERT Solutions for Class 11 Maths Chapter 16 Probability

Applications Of Probability

Let’s talk about the billion-dollar card game: Poker.

Poker is a card game that includes decision-making based on probability, psychology, and game theory. What is the probability of dealing a pair of aces in Texas Holdem Poker(the dealer dealt 2 cards to all the players)?

( 4/52 )*( 3/52 ) = (12/2652) = 0.45% (approx.)

To put this in perspective, if you’re playing poker at your local casino and are dealt 25 hands per hour, you can expect to receive pocket Aces an average of once every 9 hours.

Poker professionals use all combinations of cards and their probabilities that can help in the game. Even a small change in probability can lead them to call, raise or fold. Not only poker, in all card games, including blackjack, rummy, patience, Probability, and Statistics are also used to make the game easier.

How To Study Statistics And Probability?

There are multiple ways of studying Probability and Statistics. In this section, I have written the most preferred way of understanding probability. I recommend you to study Probability before Statistics since there are applications of probability in statistical analysis.

You can start with the Algebra of Events(Boolean Algebra). Then you can move to learn Probability using Venn diagrams. In that, you will learn the Basic Probability Formulas and their applications.

I would recommend you to study Permutations and Combinations or Combinatorics before studying Probability or start the next section after Venn diagrams. Permutations and Combinations or Combinatorics is a segment of mathematics concerned with counting. Do Combinatorics-based problems in probability. Practice till you have command of this topic as it is the most basic part of Probability. Also, practice applications of probability in other sections of mathematics.

After that, you can move to Random Variables, Conditional Probability and Expectations, and Infinite Sequences.

For statistics, Start from definitions of mean, mode, and median and then learn to mean, mode, and median for grouped and ungrouped data.

Books Recommended For Statistics And Probability

1. NCERT Book for Class 11 (Chapter 16).

2. NCERT Book for Class 12 (Chapter 13).

3. Basic Probability Theory (Robert B. Ash, Department of Mathematics, University of Illinois)

4. Head First Statistics by Griffiths.

5. Intermediate mathematical statistics Book by G. P. Beaumont

6. Introduction to the Theory of Statistics byAlexander M Mood

Maths Chapter Wise Notes For Engineering Exams

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