[Maths Class Notes] on Probability Pdf for Exam

Suppose you are playing the game of dart and aiming at the dartboard at a particular angle. A mathematician with knowledge of the game observes a few things and says that the chance of you hitting the brown space is 52%, blue space is 20%, green space is 28% and yellow space is 0%. Now, the question arises ‘on what basis did he calculate probability? And How?                  

Let us see how and understand from this article in an effective manner.

What is Probability?

It is the astounding part of Maths that deals with the outcome of a random event. The word probability means chance or possibility of an outcome. It explains the possibility of a particular event occurring. We often use sentences like – ‘It will probably rain today, ‘he will probably pass the test’, ‘there is very less probability of getting a storm tonight’, ‘most probably the price of onion will go high again. In all these sentences, we replace words like chance, doubt, maybe, likely, etc., with the word probability. Probability is basically the prediction of an event that is either based on the study of previous records or the number and type of possible outcomes.

The Story Behind the Discovery of Probability

In the 16th century, a gambler named Chevalier de Mere wanted to find out about the chances of a number appearing on the roll of dice, so he decided to approach a French Philosopher and Mathematician Blaise Pascal to solve the dice problem. Blaise Pascal got interested in the concept of possibility and so he discussed it with another French Mathematician, Pierre de Fermat. Both the Mathematicians started working on the concept of probability separately. 

Later, J. Cardan, an Italian Mathematician, wrote the first book named ‘Book on Games of Chance’ in 1663 that deals with the inception of probability. This caught the attention of some of the great Mathematicians J. Bernoulli, P. Laplace, A.A Markov and A.N.Kolmogorov.

Out of all the Mathematicians, A.N.Kolmogorov, a Russian mathematician, treated probability as a function of outcomes of the experiment. With the help of this concept, we can find the probability of events allied with discrete sample spaces. This also establishes the concept of conditional probability, which is important for the perception of Bayes’ Theorem, multiplication rule, and independence of events. In 1812, Laplace also came up with ‘Theory Analytique des Probabilities’, which is considered as the greatest contribution by an individual to the theory of probability. The deductions and reasoning introduced by these mathematicians related to probability are now being used in Biology, economics, genetics, physics, sociology, etc. 

Definition of Probability

“Probability is a mathematical term for the likelihood that something will occur. It is the ability to understand and estimate the possibility of a different combination of outcomes.” 

Probability means that it is possible. It is a branch of statistics that deals with the occurrence of a random event. The number is expressed from zero to one. Our Probability is presented in Maths to predict how likely events will occur. 

The definition of Probability is basically the degree to which something can happen. This is a basic probability theory, also used in possible dissemination, in which you will learn the feasibility of randomized test results. In order to determine the probability of a single event occurring, first of all, we need to know the total number of possible consequences.

Description of Statistical Probability

Chances are the probability of an event occurring. Many events cannot be predicted with absolute certainty. We can only predict the probability of an event occurring, that is, how likely it is that we will use it. 

Chances are from 0 to 1, where 0 means that the event does not happen and 1 indicates a particular event. 10th-grade Probability is an important student topic that explains all the basic concepts of this topic. The chances of all events in the sample space can be up to 1.

For example, if we throw a coin or find a Head or Tail, there are only two possible consequences (H, T). But if we throw two coins into the air, there may be three chances for events to occur, such as both coins showing heads or both tails or one showing heads and one tail, i.e. (H, H), (H, T). ), (T, T).

Probability Formula

The probability formula is defined as the probability of an event occurring equally to the average number of positive outcomes and the total number of outcomes.

Possibility of incident P (E) = Number of positive outcomes / Total number of outcomes. This is a basic formula. But there are other additional formulas for different situations or events.

Types of Probability

There are three main types of Probability:

1. Theoretical Probability

2. Probability to Test

3. Axiomatic Probability

Theoretical Probability

It is based on the probability that something will happen. Theoretical possibilities are primarily based on the concept of Probability. For example, if a coin is tossed, the chance of a head-turning theory will be ½.

Probability to Test

It is based on the basis of test recognition. Test scores can be calculated based on the number of possible results for the total number of tests. For example, if a coin is thrown 10 times and heads are recorded 6 times at a time, the probability of checking heads is 6/10 or 3/5.

Axiomatic Probability

With axiomatic possibilities, a set of rules or set axioms apply to all types. These axioms are set by Kolmogorov and are known as the three axioms of Kolmogorov. With the axiomatic approach to probability, the probability of occurrence or non-occurrence of events can be estimated. 

The axiomatic probability study incorporates this concept in detail with three Kolmogorov rules (axioms) and various examples.

Terms The conditions are the probability that an event or outcome will occur based on the occurrence of a previous event or outcome.

Event Probability

Assume that event E can occur in ways r without the sum of n possible or possible ways equally. Then the chances of an event or success being achieved are highlighted as;

P (E) = r / n

The chances of an event not occurring or being known as a failure are set out as follows:

P (E ’) = (n-r) / n = 1- (r / n)

E ’represents that the event will not take place.

So, now we can say;

P (E) + P (E ’) = 1

This means that the sum of all the possibilities for any random test or test is equal to 1.

What Events are Equally Possible?

When events have the same possibilities for the theatre to occur, they are called equally possible events. The results of the sample space are called equally possible if they all have the same probability of occurrence. For example, if you throw a dice, then the chances of getting 1 are 1/6. Similarly, the probability of finding all the numbers from 2,3,4,5 and 6, one at a time, is 1/6. Therefore, the following are examples of equally possible events when dice are thrown:

Finding 3 and 5 by throwing a dice

Fin
ding the same number and the odd number in the den

Find 1, 2, or 3 rolling dice

they are almost the same events, as the probability of each event is equal.

Related Events

Chances are there will be only two outcomes that the event will take place or not. As someone who will come or not come to your house, get a job or not get a job, etc., they are examples of parallel events. Basically, the completion of an event that happens exactly the opposite is likely to happen. Some examples are:

• It will rain or equal today

• The student will pass the test or not.

• You win the lottery or you don’t win.

Probability Theory

The theory that it may have had its origin in the 16th century when J. Cardan, an Italian mathematician, and physician, talks about the first work on the subject, The Book on Games of Chance. Since its inception, knowledge of Probability has brought the attention of senior mathematicians. 

Therefore, Probability theory is a mathematical component that deals with the occurrence of incidents. Although there are many different interpretations of possibilities, Probability theory directly interprets the concept by expressing it through a set of axioms or hypotheses. 

These ideas help to create Probability in terms of potential space, which allows the holding value between 0 and 1. This is known as the probability measure in a set of possible results for the sample space.

Probability Density Function 

Probability Density Function (PDF) is a probability function that represents the density of a random variance within a given range of values. The Probability Density Function describes the general distribution as well as its presence and deviation. 

The most common distribution is used to create a website or statistics, often used in science to represent real value variables, the distribution of which is unknown.

Terms related to Probability

1. Random Experiment: The accomplishment of action without any prior conscious decision results in a set of possible outcomes. This action is called the Random experiment. Probability is the prediction of a particular outcome of a random event. For example- rolling a die, tossing a coin, and drawing a card from a deck are all examples of random experiments.

2. Outcome: The result of any random experiment is called an outcome. Suppose you tossed a coin and got your head as the upper surface. So, tossing a coin is a random experiment that results in an outcome ‘head’. 

3. Sample Space: It is a set of all the possible outcomes for a random experiment. For example – Obtaining a head or a tail on the tossing of a coin is possible. Thus, the Head and Tail are the sample space. Similarly, on rolling a die, we can get either of the following numbers – 1, 2, 3, 4, 5, 6. Thus, 1, 2, 3, 4, 5, 6 are the sample space. There are six sample spaces or possible outcomes if a card is drawn from a deck.

4. Equally Likely Outcomes: When the relative occurrence of outcomes of a random experiment turns out to be equal a large number of times, then the outcomes are called equally likely outcomes. For example, the relative occurrences of Head and Tail on tossing a coin for a very large number of tosses are equal. So, Head and Tail are equally likely outcomes that make the tossing of a coin fair and unbiased if it’s to decide between two options. 

5. Event: In the case of a random experiment, an event is a set of possible outcomes at a specified condition. Example – On rolling of a die, 4 is not obtained. This event is the random experiment that is rolling of a die whose result is not 4. Thus, this event has 5 possible outcomes that are 1, 2, 3, 5, 6. Suppose it’s mentioned that the event F is equal to obtaining a black card while drawing a card from a deck. In this case, the event F has 26 possible outcomes because there are 26 black cards, all total that is 13 spades and 13 clubs. 

 

Types of Event:

1. Complementary events

2. Independent events

3. Mutually exclusive events

Types of Probability

There are three major types of probabilities:

1. Theoretical Probability- Prediction about a particular event can be precisely made with the access of statistical data of an event. The definition of probability in statistics is based on the possibility of the occurrence of an outcome. Suppose if you are willing to find out the theoretical probability of getting a number ‘5’ on rolling a die, then you should determine the number of possible outcomes. We are aware of the fact that a die has 6 numbers (i.e, 1,2,3,4,5,6), thus the number of possible outcomes is also six. So, the chance of getting 6 on rolling a die is one out of six; that is 1:6. Similarly, we know that the total number of possible outcomes on tossing a coin is 2 because you can either get your head or tail. Thus, the theoretical probability of getting head on tossing a coin is ½.

2. Experimental Probability- Experimental probability is the definition statistics of unlike theoretical probability definition includes the number of trials. Suppose a coin is tossed 30 times and out of those 30times, we got tails 12 times, then the experimental probability of getting ahead is 12:30. This calculation of probability is based on the prior carried out experiments. Experimental probability is equal to the number of all the possible outcomes of an event divided by the total number of trials. For example- a die rolled 50 times results in the appearance of 6 thrice. So the Experimental probability of getting six is 6/50. 

3. Axiomatic Probability- Axiomatic Probability is a theory of unifying probability where there is an application of a set of rules made by Kolmogorov. 

The three axioms are:

1. The probability of an event A is always greater or equal to zero but can never be less than zero.

2. If S is a sample space, then the probability of occurrence of sample space is always 1. That is, if the experiment is performed, then it is sure to get one of the sample spaces.

3. For mutually exclusive events, the probability of either of the events happening is the sum of the probability of both the events happening.

Formula for Probability

When the possibility of occurrence of each outcome is the same in a particular event, the experiment or event is said to have equally likely outcomes. For example, on rolling
a die, the possibility of getting a number is equally likely but getting a red ball from the bag of four red balls and 2 blue balls is not equally likely.

On the basis of the experimental formula, we can say that the probability is:

On the basis of the theoretical formula, we can say that the probability is:

 [P(E) = frac{text{Number of trails in which the event happened}}{text{Total number of trails}}]

On the basis of the theoretical formula, we can say that the probability is:

[P(E) = frac{text{number of outcomes favorable to E}}{text{Number of all possible outcomes of the experiments}}]

Example 1: What is the probability of getting a tail if a coin is tossed once?

Solution: The total number of possible outcomes is 2 that is Head and Tail.

Let the event of getting a tail be E.

The probability of getting a tail on tossing a coin is:  

[P(E) = frac{text{Number of outcomes favorable to E}}{text{Number of all possible outcomes of the experiments}}]

=[ frac {1}{2}]

Example 2: A bag contains a blue ball and a red ball and a yellow ball of the same size and weight. If Archana picks out a ball from the bag randomly, then what is the probability of getting an (i) blue ball (ii) yellow ball, and (iii) red ball. 

Solution:

The total number of balls inside the bag is 3 out of which one ball is red, one ball is blue, and yellow. If Archana takes out a ball from the bag randomly then 

(i) The probability of getting a blue ball = 1/3

(ii) The probability of getting a yellow ball = 1/3

(iii) The probability of getting a red ball = 1/3

Uses of Probability

Probability is important to figure out if a particular thing is going to occur in an event or not. It also helps us to predict future events and take action accordingly. Below are the uses of probability in our day-to-day life.

1. Weather Forecasting- We often check weather forecasting before planning for an outing. The weather forecast tells us if the day will be cloudy, sunny, stormy or rainy. On the basis of the prediction made, we plan our day. Suppose the weather forecast says there is a 75% chance of rain. Now, the question arises how is the calculation of probability or precise prediction done. The access to the historical database and the use of certain tools and techniques helps in calculating the probability. For example, according to the database, if out of 100 days, 60 days were cloudy, then we can say that there is a 60% chance that the day will be cloudy depending on other parameters like temperature, humidity, pressure, etc.

2. Agriculture- Temperature, season, and weather play an important role in agriculture and farming. Earlier, we did not have a better understanding of weather forecasting, but now various technologies are developed for weather forecasting, which helps the farmers to do their job well on the basis of predictions. Undoubtedly, the occurrence of erratic weather is beyond human control, but it is possible to prepare for the adverse weather if it is forecasted beforehand. The process of sowing is usually done in clear weather. Thus, the accurate prediction of weather enables the farmer to take major steps in order to prevent big losses by saving their crops. The planning of other suitable farming operations like irrigations, application of fertilizers and pesticides, etc., depends on the weather. Thus a proper weather forecast is needed.

3. Politics- Many politicians want to predict the outcome of an election even before the polling is done. Sometimes they predict which political party will rise to power by closely studying the results of exit polls. There are some politicians who spend a lot only to predict the results so that they can save themselves from being dethroned. There are other good uses of probability, like predicting the number of students who would be needing jobs in the upcoming year so that the vacancy can be created accordingly. Politicians can also analyze the rate of car and bike accidents increased in past years so that they can take measures and reduce road accidents.  

4. Insurance- Insurance companies use probability to find out the chances of a person’s death by studying the database of the person’s family history and personal habits like drinking and smoking. Probability also helps to examine and evaluate the best insurance plan for the benefit of a person and his family. Suppose a person who is an active smoker has more chances of getting lung cancer as compared to the people who don’t. Thus, it is beneficial for a smoker to go for health insurance rather than vehicle or house insurance for the betterment of his family.