Conditional Probability

The concept of Conditional Probability forms the basis of Bayes Theorem. Bayes Theorem forms the concept of Naive Bayes Classifier.

A simple formulation of Conditional Probability

How do we read this above equation…?

Suppose we have two events A and B.
Finding out the probability of A, given that event B has already occurred.
{The former part is the condition here.}
Probability of A intersection B or we can say Probability of A and B
divided by probability of B.
The denominator is generally the one which serves as the condition.

Let’s understand this using a Venn diagram,

When we don’t have any condition, our probability is the entire sample space or the entire universe.
The moment we add condition, our sample space gets reduced and we have a new sample space.
On the basis of the new sample space which gets generated, we will calculate our conditional probability.
Going back to finding P( A | B ), which mean probability of finding the occurrence of event A given that B has already occurred, we can represent it this way,

So, initially we had the entire universe as our sample space, as we add a new condition our sample space has now shrunk to only within B. And this becomes our new sample space.
Considering this sample space we need to find our probability, the remaining part which has not been shaded will not be considered as our sample space.
Now, our sample space includes pure event B and A intersection B part.
The same parts were included in our formulae also.
Our area of consideration for our sample space is composed of both of these events.
Using these events in the formulae will help us find the probability that we interested in.

To conclude we can say that, in Conditional Probability our sample space gets reduced to a new area based on the condition provided which acts as a new sample space.

Consider another example, where we have 100 students out of which we have,

A : students who like only apple : 40
B : students who like only orange : 30
A ⋂ B : students who like both apple and orange : 20

To find : What is the probability of students liking apples given that students already like oranges?

Considering the formulae,

0.67 is the probability of students liking apples given that students already like oranges.