Bayes’ Theorem Also known as Bayes’ Rule, it is a fundamental theorem in probability theory that deals with the calculation of conditional probabilities. It is named after Thomas Bayes, an 18th-century British statistician, who first described it in his essay An Essay towards solving a Problem in the Doctrine of Chances. The rule states that for any two events A and B, the probability of A given B can be determined if we know the probability of B given A, and the probabilities of both A and B independently.

In other words, Bayes’ Rule can be used to calculate the probability of an event occurring given another event has taken place. A common way to express Bayes’ Rule mathematically is as follows:

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

This equation states that the conditional probability of event A occurring given event B has occurred is equal to the product of probabilities of events B and A separately divided by the probability of event B occurring.

### Bayes’ Rule Working

To illustrate how Bayes’ Rule works, let’s take a simple example where we are trying to determine whether or not a coin flip results in heads or tails. If we assume that there is an equal chance for either outcome (i.e., 50%) then we can use Bayes’ Rule to calculate our chances if we know one result from prior flips: Let’s say we flipped a coin four times, and three out of those four flips resulted in heads – what would be our chances that the fifth flip would result in heads? Using Bayes’ Rule, this would be calculated as (3/4)*(1/2)/(4/4), which simplifies to 3/8 or 37.5%. This means that with three out of four previous flips resulting in heads, our chances for getting a head on the fifth flip are slightly better than if we had no prior information about outcomes; now they stand at 37.5%, rather than 50%.

Bayes’ Rule enables us to make predictions based on knowledge from prior data while also taking into account any new evidence that might change our beliefs regarding future events – which could prove invaluable when making decisions involving uncertainty or risk. Furthermore, it allows us to estimate probabilities even when complete information isn’t available. As such, it has been used extensively across fields such as Mathematics, Statistics, Machine Learning and Artificial Intelligence and increasingly finds applications in areas such as Natural Language Processing, Computer Vision, and Medical Diagnostics.