Probability is a fundamental concept in mathematics that plays a crucial role in various fields such as statistics, economics, physics, and computer science. Understanding probability allows a person to make informed decisions, predict outcomes, and analyse uncertainties.

What is Probability?

At its core, probability is the measure of the likelihood that an event will occur. This measure is expressed as a number between 0 and 1, where 0 indicates that the event will not occur, and 1 indicates that the event will definitely occur. For example, the probability of flipping a fair coin and getting heads is 0.5 because there are two equally likely outcomes (heads or tails).

Basic Concepts

Sample Space and Events

In probability theory, lets begin by defining a sample space.

The sample space, denoted by S, is the set of all possible outcomes. For example, when rolling a fair six-sided die, the sample space consists of the numbers {1, 2, 3, 4, 5, 6}, so S={1,2,3,4,5,6}. When you roll a six-sided die, each side has an equal chance of landing face-up. For a six-sided die, each number represents one possible outcome of a roll.

An event (let’s denote this event by E), is any subset of the sample space, representing one or more outcomes of interest. For instance, rolling an even number (2, 4, or 6) is an event. So, E={2,4,6}.

The probability of an event happening is the ratio of the number of favourable outcomes to the total number of outcomes in the sample space.

P(E)=Number of outcomes in E/Total number of outcomes in S.

For rolling an even number: P(E)=3/6=1/2​

Since there are 3 even numbers (2, 4, and 6) out of 6 possible outcomes.

Complementary Events

The complement of an event E, denoted E′ or Eˉ, represents all outcomes not in E. For example, if E is the event of rolling an even number, then E′ is the event of rolling an odd number. In this case E′={1,3,5}.

Independent Events

Events are independent if the occurrence of one event doesn’t affect the occurrence of another. For instance, if you roll the die twice, the outcome of the first roll doesn’t affect the outcome of the second roll.

Dependent Events

Events are dependent if the occurrence of one event affects the occurrence of another. For example, if you draw a card from a deck and don’t replace it, the probability of drawing a particular card on the second draw changes because there is one less card in the deck.

Types of Probability

Classical Probability

Classical probability, also known as theoretical or a priori probability, relies on equally likely outcomes. It is applicable in situations where all outcomes are known and have the same chance of occurring. For example, when rolling a fair six-sided die, each face has an equal probability of 1/6​.

Empirical Probability

Empirical probability, also called experimental or relative frequency probability, is based on observed outcomes from experiments or real-life data. It involves conducting trials and recording the frequency of occurrence of an event. For instance, the probability of it raining tomorrow can be estimated by analysing historical weather data.

Subjective Probability

Subjective probability is based on personal judgment or beliefs about the likelihood of an event occurring. It is often used when there is insufficient data or when the outcomes are subjective. For example, estimating the probability of a football team winning a match based on one’s knowledge of the team’s performance and the opponent’s strengths.

navigation through uncertainty

Probability is a powerful tool for analysing uncertainty and making informed decisions in various fields. By understanding the basic concepts and types of probability, you can develop a solid foundation for more advanced studies in probability theory, statistics, and other related disciplines. Whether you’re predicting the outcome of a simple coin toss or assessing the risk in a complex financial investment, probability is a versatile tool that empowers you to navigate the uncertain world around us.