Bayes Theorem Formula & Proof Bayes Theorem

What is Bayes Theorem?

Bayes theorem is a wonderful choice to find out the conditional probability. You must have heard of the Conditional Probability of an event occurs that some definite relationship with other events. For example, every time you park a car to the busiest place then the probability of getting space depends on the time you choose to park your car. In the same sense, Bayes theorem is little more nuanced. In brief, it will give you a perfect idea of the actual probability of an event or accurate details about tests.

  • Events are slightly different from tests. For example, when you are asked to perform a liver functioning test then this is different from the event actually having the liver disease.
  • If your test results are positive then it does not signify that you are actually suffering from the disease. Most of the times, the tests have a positive false rate. Please don’t relate everything to medical tests only but it could be related to other events as well.

Bayes Theorem Formula

Bayes theorem also popular as the Bayes rule, using a simple formula to calculate the conditional probability. This theorem was named after the name of popular English mathematician Thomas Bayes (1701-1761). The formula for Bayes theorem in mathematics is given as –

\[\ P(B|A) = \frac{P(A|B) P(B)}{P(A)}\]

\[\ P(A|B) = \frac{P(B|A) P(A)}{P(B)}\]

P(A/B) is the probability of A if we already know that B has occurred and is known as likelihood.
P(B) is known as prior probability and P(B/A)is posterior probability.

In most of the cases, this is easy to identify the tests and events for an equation. It may be a little bit tricky learning these concepts and understanding their technical working. Still, you should switch your tests and events around that may be a little bit confusing sometimes. A depth learning of Bayes Theorem will give you a perfect idea of how can solve the typical maths problems based on Bayes’ Theorem.

Bayes Theorem Proof

One more way to look at the Bayes Theorem is how one event follows the another. First, we discussed the Bayes theorem based on the concept of tests and events. Now its time to study Bayes theorem in form of terms where the first term leads to the second term.

Take an example of the clinic where 10 percent of patients are prescribed with narcotic painkillers. Almost 5 percent are addicted to narcotic drugs. The patients that are prescribed with painkillers, 8 percent are addicted. If the patient is an addict then what is the probability that they will be prescribed with painkillers.

  • Step 1 – Here, first note down the percentage of people prescribed with pain pills i.e. 10%
  • Step 2 – Note down the people who are addict i.e. given 5%.
  • Step 3 – Now calculate the probability of event B with respect to the given event A. In brief, you need to find the (B|A). We want to know “Given that people are prescribed pain pills, what’s the probability they are an addict?” That is given in the question as 8%.
  • Step 4 – Now you just have to put the values from Step 1, 2 and 3 into the formula and find the answer.