How does Bayes’ Theorem work?

Bayes’ Theorem works by updating our belief or probability of an event occurring based on new information or evidence that becomes available.

To understand how it works, let’s consider an example:

Suppose we have two boxes, Box A and Box B. Box A contains 3 red balls and 2 green balls, while Box B contains 2 red balls and 3 green balls. We randomly choose one box, and then randomly choose one ball from the box.

Now, let’s say we know that the ball we picked is green. We want to know the probability that the ball came from Box A.

Using Bayes’ Theorem, we can calculate this probability as follows:

P(Box A | Green Ball) = P(Green Ball | Box A) * P(Box A) / P(Green Ball)

where: P(Box A | Green Ball) is the probability that the ball came from Box A given that it is green. P(Green Ball | Box A) is the probability that we pick a green ball from Box A. P(Box A) is the prior probability that we picked Box A. P(Green Ball) is the overall probability of picking a green ball (which can be calculated by adding the probability of picking a green ball from Box A to the probability of picking a green ball from Box B).

Using the information we have, we can calculate the probabilities as follows:

P(Green Ball | Box A) = 2/5 P(Green Ball | Box B) = 3/5 P(Box A) = 1/2 P(Green Ball) = (2/5 * 1/2) + (3/5 * 1/2) = 1/2

Plugging these values into the formula, we get:

P(Box A | Green Ball) = (2/5 * 1/2) / (1/2) = 2/5

So the probability that the ball came from Box A given that it is green is 2/5 or 40%.

In this example, Bayes’ Theorem allowed us to update our prior probability based on new information (the fact that the ball we picked is green) and calculate the probability of the event we are interested in (the ball came from Box A).