Conditional Probability and Bayes’ Theorem
Bayes’ Theorem is a fundamental concept in probability theory named after the Reverend Thomas Bayes. It provides a mathematical framework for updating beliefs or hypotheses in light of new evidence or information. This theorem is extensively used in various fields, including statistics, machine learning, and artificial intelligence.
At its core, Bayes’ Theorem enables us to calculate the probability of a hypothesis being true given observed evidence. The theorem is expressed mathematically as follows:
P(A∣B) = (P(B∣A) × P(A)) / P(B)
Where:
- P(A∣B) is the posterior probability of hypothesis A given evidence B.
- P(B∣A) is the likelihood of observing evidence B given that hypothesis A is true.
- P(A) is the prior probability of hypothesis A before observing any evidence.
- P(B) is the probability of observing evidence B regardless of the truth of hypothesis A.
Here’s a breakdown of how Bayes’ Theorem works:
- Prior Probability P(A): This represents our initial belief in the likelihood of hypothesis A being true before considering any new evidence.
- Likelihood P(B∣A): This indicates the probability of observing the evidence B given that hypothesis A is true. It quantifies how well the evidence supports the hypothesis.
- Evidence P(B): This term serves as a normalization factor and represents the total probability of observing the evidence B across all possible hypotheses.
- Posterior Probability P(A∣B): This is the updated probability of hypothesis A being true after taking into account the observed evidence B. It’s what we’re ultimately interested in determining.
Bayes’ Theorem is particularly powerful because it allows us to incorporate new evidence incrementally, refining our beliefs as more data becomes available. This iterative process of updating beliefs with new evidence forms the basis of Bayesian inference, which is widely used in fields such as medical diagnosis, spam filtering, weather forecasting, and many others.
Bayes’ Theorem provides a principled approach for reasoning under uncertainty, making it a cornerstone of probabilistic reasoning and decision-making in diverse domains.
Conditional Probability
Conditional probability is one type of probability in which the possibility of an event depends upon the existence of a previous event. As this type of event is very common in real life, conditional probability is often used to determine the probability of such cases.
Conditional probability describes the likelihood of an event (A) happening given that another event (B) has already occurred. In probability notation, this is denoted as A given B, expressed as P(A|B), indicating that the probability of event A is dependent on the occurrence of event B.
To know about conditional probability, we need to be familiar with independent events and dependent events. Let’s understand conditional probability, and its formula with solved examples in this article.
Table of Content
- What is Conditional Probability?
- Conditional Probability Definition
- Conditional Probability Formula
- How to Calculate Conditional Probability?
- Conditional Probability of Independent Events
- Conditional Probability vs Joint Probability vs Marginal Probability
- Conditional Probability and Bayes’ Theorem
- Conditional Probability Examples
- Tossing a Coin
- Drawing Cards
- Properties of Conditional Probability
- Multiplication Rule of Probability
- How to Apply the Multiplication Rule?
- Applications of Conditional Probability
- Conditional Probability Questions