What is Probability? The Ultimate Guide

'Uncertainty' is a word that fills us all with dread, and often, we are bound to make day-to-day decisions based on mere chance. But what if there's a way to narrow down our chances into the realm of certainty? What if you could bridge the gap between uncertainty and possibility with confidence? The mathematical field of Probability helps you achieve that with great precision. Understanding What is Probability is not only helpful; it’s essential to navigate this world of fleeting chances and possibilities.

From everyday scenarios such as weather predictions to more complex applications in science, politics and finance, Probability offers a precise glimpse of the most likely outcomes. This blog dissects What is Probability in detail covering its key terminologies, formulas, theorems and more. So read on, take a chance at mastering the art and science of Probability, and become a pro at prediction! 

Table of Contents

1) What is Probability in Maths? 

2) Events in Probability  

3) Probability Formulas 

4) Different Types of Probability 

5) Probability Theorems 

6) Examples of Probability 

7) Applications of Probability 

8) How to Figure out the Odds of Something?

9) How to Calculate the Probability of Something Not Happening?

10) Conclusion 

What is Probability in Maths?

Probability defines the likelihood of an event's occurrence. There are countless real-life situations which may compel us to predict the outcome of an event, but we may not be sure of the result. In such cases, this event is said to be probable to occur or not occur. Probability boasts great applications in business and games to make predictions, with extensive applications in this field of Artificial Intelligence (AI).

Key Terms in Probability 

Here are some foundational concepts or terminologies used in Probability: 

1) Sample Space: This is the set of all possible outcomes in a Probability experiment. For instance, in the case of a coin toss, it’s “heads” and “tails”. 

2) Sample Point: This defines one of the possible results of an experiment. For example, sample points are 1 to 6 when rolling a fair six-sided dice. 

3) Experiment: This refers to a process with uncertain results. Examples include card selection, coin tossing, or rolling a die. 

4) Event: This defines a subset of the sample space representing specific outcomes, such as getting “3” when rolling a die. 

5) Favourable Outcome: This outcome produces the desired or expected consequence. 

 

 

Events in Probability 

According to Probability theory, an event is a set of outcomes of an experiment. Let's say P(E) represents the Probability of event E, then we get:

1) P(E) = 0 if and only if E is an impossible event

2) P(E) = 1 if and only if E is a certain event

3) 0 ≤ P(E) ≤ 1

Let's say we have two events, A and B. The Probability of event A, P(A) > P(B) if and only if event A is likelier to occur than event B. Sample space(S) is the set of all of possible outcomes of an experiment and n(S) is the number of outcomes in the sample space.

E’ indicates that the event won’t occur.

Therefore, we can also conclude that, P(E) + P(E’) = 1

Probability Formulas 

The Probability formulas for the events A and B are summarised below: 

Different Types of Probability 

There are three major types of probabilities, namely Theoretical Probability, Experimental Probability and Axiomatic Probability. These are explored in detail below 

Theoretical Probability 

Theoretical Probability is where the likelihood of something occurring is determined without the assistance of an experiment. The Probability of an event is reached through prior experiments.  It's represented in Maths in this way:

Experimental Probability 

Experimental Probability (or Empirical Probability) is the ratio between the number of trials carried out and the number of times a particular event occurs. Experimental Probability matches the general definition of Probability but within the context of a research study or experiment.

To determine this, researchers usually experiment multiple times to obtain trustworthy results. Mathematically, Experimental Probability is represented as follows:

Axiomatic Probability 

In case of Axiomatic Probability, a set of rules or axioms are applied to every type. The chances of occurrence (or non-occurrence) of an event is quantified by applying these axioms as explained below: a) The smallest possible Probability is zero, and the biggest is one. 

b) A certain event has a Probability equal to one. 

c) Any two mutually exclusive events can't occur simultaneously, while the union of events says only one can occur. 

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Probability Theorems 

The following Probability Theorems find widespread application in Mathematics:

1) Addition Theorem: If A and B are two events, the Probability of their union is:

2) Multiplication Theorem: This describes the Probability of the intersection of two events A and B:

3) Law of Total Probability: If {B1,B2,…,Bn} is a partition of the sample space, the Probability of an event A is:

4) Bayes' Theorem: This describes the Probability of an event’s occurrence related to any condition. The formula for Baye’s Theorem is:

5) Complement Rule: The Probability of the complement of an event A is:

6) Independence Rule: Two events A and B are independent if:

7) Conditional Probability Formula: The Probability of event A under the condition that B has occurred is:

8) Bernoulli’s Theorem (Law of Large Numbers): As the number of independent trials increases, the relative frequency of an event converges to its true Probability. This theorem is expressed as follows:

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Examples of Probability 

Here are a couple of examples to illustrate the concept of Probability 

Applications of Probability 

Probability boasts a broad variety of applications in real life. Some basic applications include the following events:

1) Choosing a card from the deck of cards

2) Flipping a coin

3) Throwing a dice in the air

4) Winning a lucky draw

Major applications of Probability include the following:

1) It is used for Risk Assessment and modelling in diverse industries.

2) It's employed in weather forecasting or prediction of weather changes.

3) It is used to calculate the chances of a team winning in a sport depending on the players and team strength.

4) In the Share Market, Probability is crucial in determining the chances of getting a hike of share prices.

How to Figure out the Odds of Something? 

You can first calculate the Probability and then convert it to odds by dividing the Probability by one minus that Probability. So, if the Probability is 20% or 0.20, the odds are 0.2/0.8 or ‘2 to 8’ or 0.25.

How to Calculate the Probability of Something not Happening?

You can use the following formula to calculate the Probability of something not happening:

P(A') = 1−P(A)

Here, P(A) = Probability of the event occurring. 

P(A') = Probability of the event not occurring.

Conclusion 

In conclusion, mastering Probability is a powerful way to understand the uncertainties we experience on a daily basis. More than just numbers, having a firm grasp on What is Probability is about embracing the unpredictable facets of life. Whether you’re rolling a dice or calculating risks for your business, this blog's exploration of Probability will guide you to handle uncertainties with precision. 

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