Probability in mathematics deals with assessing the likelihood of events, essentially informing us about the possibility of their occurrence. This probability value invariably ranges between 0 and 1, offering insights into the certainty or uncertainty of an event taking place. Probability becomes invaluable when we encounter situations where we’re uncertain about the outcome of an event. In this article, we will delve into Probability, encompassing its definition, formula, associated terminology, and various types of probability, as well as an exploration of Bayes’ Theorem and related concepts.

Defining Probability in Mathematics

In mathematics, probability serves as a tool for assessing the likelihood of events. It offers a means to gauge the likelihood of occurrences that are often challenging to predict with absolute certainty. Probability values, which fall within the range of 0 to 1, help us express the degree of improbability (0) or certainty (1) associated with an event. Calculating probability involves a straightforward process of dividing the number of favorable outcomes by the total number of potential outcomes.

Terminology of Probability Theory 

Sample Space:

• The sample space is a fundamental concept in probability theory. It encompasses all possible outcomes of a random experiment or trial. For example, when flipping a coin, the sample space consists of two outcomes: “head” and “tail.” It represents the entire universe of possible results for that specific experiment.

Sample Point:

• A sample point, sometimes referred to as an elementary event, is a single outcome within the sample space. In the context of rolling a fair six-sided die, the sample space contains six possible sample points, which are the individual numbers from 1 to 6: 1, 2, 3, 4, 5, and 6.

Experiment:

• An experiment is a process or trial in which we observe or measure some uncertain or random phenomenon. This could be as simple as tossing a coin, selecting a card from a deck, or rolling a die. In essence, any situation where the results are uncertain or unpredictable can be considered an experiment in probability.

Event:

• An event is a subset of the sample space. It represents a specific outcome or a set of outcomes that you are interested in. For example, when rolling a six-sided die, the event of getting an even number (2, 4, or 6) is a subset of the sample space, and it’s considered an event.

Favorable Outcome:

• A favorable outcome is an outcome within an event that leads to the desired or expected result. For instance, if you’re interested in the event of rolling a 1 with a six-sided die, the outcome of rolling a 1 is the favorable outcome for that event.

Equally Likely Events:

• Equally likely events are events with the same probability of occurring. In other words, each event within this category has an equal chance of happening. For instance, when flipping a fair coin, getting either a “head” or a “tail” is equally likely, as both have a 50% probability.

Exhaustive Events:

• An event is considered exhaustive when it includes all possible outcomes within the sample space. This means that if you list all the events, together they should cover every possible outcome of the experiment. In essence, the events collectively form the sample space.

Mutually Exclusive Events:

• Mutually exclusive events are events that cannot happen at the same time. If one of these events occurs, it prevents the occurrence of the other. For example, in a weather forecast, “hot” and “chilly” weather are mutually exclusive because both cannot happen simultaneously.

Complementary Events:

• Complementary events are pairs of events where one represents the occurrence of an event, and the other represents the non-occurrence or opposite of that event. The complement of event A, denoted as A’ or “not A,” includes all outcomes that are not in event A. For example, if event A is getting a head when flipping a coin, its complement (A’) is getting a tail. Complementary events cover all possibilities, so the sum of their probabilities equals 1.

Probability Density Function

A probability density function (PDF) is a fundamental concept in probability theory used to determine the probability of a continuous random variable falling within a specific range of values rather than any single value. It is a non-negative function, and the area under the PDF curve always equals one. While the PDF provides the likelihood of values of the continuous random variable, it’s important to note that the PDF value itself doesn’t represent the probability of a particular X-value; it denotes density.

For continuous random variables, the probability at a specific point is 0, and to calculate probabilities, integration over a defined interval is necessary, with the length of the interval affecting the probability, which is always less than 1 within that range. In essence, the PDF characterizes the likelihood of a continuous variable’s values within a range.

Probability Formula

The probability formula quantifies the likelihood of an event occurring and is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. It can be expressed as:

P(E) = Number of Favorable Outcomes / Total Number of Outcomes

• P(E) represents the probability of an event denoted as “E.” This is the value you are trying to calculate, which indicates the likelihood of event E happening.
• Number of Favorable Outcomes refers to the count of outcomes that correspond to the event of interest, the outcomes you want to happen or observe.
• Total Number of Outcomes represents the entire set of possible outcomes for the particular experiment or situation. It encompasses all potential results, including both the desired outcomes (favorable) and those that are not (unfavorable).

Probability Tree Diagram

A probability tree diagram visually organizes information for both dependent and independent events. It displays all possible outcomes, typically beginning with the initial event and probabilities on branches. To find the probability of a specific event, multiply the branch probabilities. The sum of all branch probabilities equals 1. To create one, identify event dependencies, draw branches for outcomes, add probabilities, and extend for subsequent events. Probability tree diagrams are useful for solving complex probability problems involving multiple stages, helping calculate the likelihood of different outcomes.

Bayes’ Theorem on Conditional Probability 

Bayes’ Theorem is a mathematical formula designed to compute conditional probabilities, specifically, the probability of an event happening given that another event has already occurred. The formula is expressed as:

P(A|B) = (P(B|A) * P(A)) / P(B)

Here’s the breakdown:

• P(A|B) is the probability of event A occurring given that event B has already happened.
• P(B|A) is the probability of event B occurring given that event A has already occurred.
• P(A) is the prior probability of event A occurring.
• P(B) is the prior probability of event B occurring.

Bayes’ Theorem finds applications in diverse fields such as medical diagnosis, spam filtering, and situations where our beliefs need updating based on new information. For instance, if a patient tests positive for a rare disease, we can use Bayes’ Theorem to calculate the probability that the patient genuinely has the disease, considering the test’s accuracy and the disease’s prevalence in the population.

FAQs on Probability in Maths

Q1. What is Probability? Give an example.

• Probability is a mathematical concept that measures the likelihood of an event occurring. For example, when tossing a coin, there are two possible outcomes: heads or tails.

Q2. How many types of Probability are there?

• There are three main types of probability: Theoretical Probability, Experimental Probability, and Axiomatic Probability.

Q3. What is the formula for Probability?

• The probability formula is: P(E) = Number of favorable outcomes / Total number of outcomes, where P(E) represents the probability of event E.

Q4. What does a Probability of 0 mean?

• A probability of 0 signifies that the event does not occur.

Q5. What is the Probability Formula?

• The probability formula calculates the likelihood of an event by dividing the number of favorable outcomes by the total possible outcomes.

Q6. How To Calculate Probability?

• To calculate probability, find the ratio of favorable outcomes to the total outcomes in the sample space. The formula is: Probability of an event P(E) = (Number of favorable outcomes) ÷ (Number of Elements in Sample space).