In mathematics, probability is a quantification of the uncertainty we face in various situations. The study of probability extends beyond mathematics and statistics as it has practical applications in different fields, including finance, medicine, and physics, to name a few. In this article, we will discuss different approaches to modelling probability, including their definitions, examples, advantages, and disadvantages.
Classical Probability
Classical probability is a theoretical approach to modelling probability. It uses a set of assumptions to define the probability of an event. In this approach, all events are equally likely to occur, and the probability of an event is calculated by dividing the number of favorable outcomes by the total possible outcomes. Some examples of classical probability include coin tosses and rolling dice. Though it has limitations in the real world, classical probability has theoretical and practical applications in various fields.
Examples of classical probability
- Flipping a coin and getting heads or tail is an example of classical probability. The probability of getting a head or tail is ½.
- Rolling a six-sided dice and getting a specific number is another example of classical probability. The probability of getting any number is ⅙.
Empirical Probability
Empirical probability, also known as experimental probability, uses the observed frequency of events to calculate the probability of an event occurring. Unlike classical probability, it is based on real-world observations and uses evidence to quantify the uncertainty. Empirical probability is useful in situations where it is difficult or impossible to calculate theoretical probabilities.
Examples of empirical probability
- Conducting a survey to find out what percentage of people prefer a particular brand of product.
- Counting the number of cars that pass by a specific site to estimate traffic.
Subjective Probability
Subjective probability is based on personal beliefs, experiences, and opinions, which results in different people having different estimations of the probability of a particular event. This method of modelling probability relies on the subjective evaluation of an event. It is useful when data is non-existent or when a situation is uncertain. However, subjective probability is prone to individual biases, which might adversely impact the accuracy of predictions.
Examples of subjective probability
- Estimating the probability of a politician winning a presidential election.
- Predicting the likelihood of a particular sports team winning a game based on their current performance and players.
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred. It is useful in situations where the outcome of one event is dependent on another event. It provides us with the likelihood of future events based on historical data.
Examples of conditional probability
- The probability of a borrower defaulting on a loan given that they have a poor credit score.
- The probability of an email being spam given that it contains certain words.
Bayesian Probability
Bayesian probability is a statistical approach to modelling probability that involves making an assumption and updating it with new data continually. It is useful in complex situations, such as decision-making in the stock market, by providing continuous adjustments in response to emerging patterns.
Examples of Bayesian probability
- Assessing the probability of an underdog winning a sporting event based on the odds and performance.
- Assessing the likelihood of a disease outbreak based on the symptoms and the frequency of similar outbreaks.
Markov Chain
A Markov chain is a mathematical model used to describe a sequence of events in which the probability of each event depends only on the state attained in the previous event. It assumes that the future is only dependent on the present state and not the past. Markov chains are used in many fields, including economics, natural language processing, and genetics.
Examples of Markov chains
- Assessing the probability of a sports team winning future games based on their recent performance.
- Predicting the likelihood of a person converting to a different religion based on their current beliefs and practices.
Conclusion
Probability is a complex concept that is used to describe uncertainty in different situations. In this article, we discussed various approaches to modelling probability, including classical, empirical, subjective, conditional, Bayesian, and Markov chains. By understanding the different models, we can make better-informed decisions in uncertain situations.
FAQs
Q.What is the difference between classical and empirical probability?
Classical probability is a theoretical model based on a set of assumptions, while empirical probability is based on real-world observations.
Q.What is conditional probability?
Conditional probability is the probability of an event occurring given that another event has already occurred.
Q.What are the advantages and disadvantages of using Bayesian probability?
Bayesian probability provides continuous updates based on new data, making it useful in complex and uncertain situations. However, it is dependent on the initial assumption and can be impacted by biases or incomplete data.
Q.How are Markov chains used to model probability?
A Markov chain provides a model to describe a sequence of events in which the probability of each event depends only on the state attained in the previous event.
Q.Can you give an example of subjective probability?
Estimating the likelihood of a particular sports team winning a game based on their current performance and players.
