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

Chance is the occurrence of events in the absence of any obvious intention or cause. It is, simply, the possibility of something happening. When the chance is defined in Mathematics, it is called probability.

Probability is the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the whole number of cases possible.

Mathematically, the probability of an event occurring is equal to the ratio of a number of cases favourable to a particular event to the number of all possible cases.

The theoretical probability of an event is denoted as P(E).  

[P(E) = frac{text{Number of Outcomes Favourable to E}}{text{Number of all Possible Outcomes of the Experiment}}]

Assume that we take a coin and toss it, the chances it lands a head is equal to the chances that it lands a tail. Similarly, in any such event, there are equal chances for any of the different cases occurring. For example, in rolling dice, all six numbers are equally likely to be obtained. We also assume that the dice or coin we use is not biased and is fair, i.e. it has not tampered with an intention to favour a particular outcome.

All the possible outcomes of a random experiment when put together is called a sample space. Each possible result of an event is called an outcome. If an event has only one outcome, then it is called an elementary event. 

The sum of all the probabilities of each elementary event is 1. For example, while tossing a coin the two possible outcomes are heads or tails.

P(getting head) =   ½

P(getting tail) = ½

Now the sum of all possible outcomes will include the probability of heads and probability of tail.

P (tossing a coin) = P (getting head) + P (getting tail)        

       =      ½ + ½ 

       =        1

Probability, in simple terms, is defined as the likelihood of a particular event to happen in the near future. It can simply be termed as a possibility. Probability lies between 0 and 1; where 0 denotes impossibility and 1 denotes surety. 

Probability has a wide variety of applications and is used widely in the fields of statistics, Mathematical branch, science, gambling, philosophy, computer science,etc. 

This particular theory also supports underlying mechanics or regularities of complex systems.

Interpretation of Probability

Probabilities can be simply described as the number of desired outcomes divided by the total number of all outcomes.

In order to understand the term, let us consider an example. For instance, you are tossing a coin. The probable events to occur are “head-head”, “tail-tail”, “head- tail”, “tail- head”.

 In terms of probability, it can be written as 1 out of 4 outcomes and in the numerical form, it is written as 1/4 which is 25% or 0.25.

 The practical applications of the theory can be majorly classified into two types:

The numbers which are assigned to describe the objectives or physical states of affairs are termed objectivists.

The numbers which are assigned to describe the subjective probability (degree of belief) are termed, subjectivists.

Applications of Probability

Probabilities play a vital role in daily life. It has a wide variety of applications. From the wide variety of probabilities applications, here are some of the most common applications listed below:

• The insurance industry and markets, in order to determine the pricing and making of trading decisions, use the Mathematical branch of study which is termed as probability.

• It is also used for governmental purposes such as in fields like environmental regulation, entitlement analysis and financial regulation.

• Probability is also used in research trends and square punnets also.

• It is used as a statistical tool in order to calculate the likelihood of undesirable events that would occur in the near future.

• Many of the consumer products which includes automobiles and consumer electronics also used the reliability theory which is a part of probability. It also helps out in predicting the failure chances, that is, the theory helps to reduce the probability of failure in product design.

• The different types of language models which are natural language processings also proves to be an effective application of probability theory which includes the cache language model or other statistical language models.

Chance and Probability are very similar to each other. Both of them have the numbers 0 and 1. The difference they share is that chance doesn’t have any obviousness whereas probability exactly defines the ratio of how likely an event is to happen.