# Developing an Intuition for Conditional Probability

Conditional probability is usually talked about in the form, “what is the probability of event A happening, given that we know B happened?”

The formula for conditional probability is:

we divide by P(B) because we are interested in the probability of A occurring given that B has already occurred. This division serves to normalize the probability of the intersection of A and B with respect to the probability of B.

To understand this intuitively, let’s break it down:

**Intersection P(A∩B)**: This is the joint probability that both A and B occur. It represents the portion of the sample space where both events happen.**Given B**: When we say “given B,” we are essentially restricting our consideration to the scenario where B has definitely occurred. The sample space is now reduced to only those outcomes where B happens.**Normalization with P(B)**: We need to adjust the joint probability P(A∩B)to reflect this restricted sample space. Dividing by P(B) normalizes the probability, effectively telling us, “Out of all the ways B can happen, how many of those also result in A happening?”

Note that the probability of A given B does NOT equal the probability of B given A.

That formula would look like this:

# Analogy

Imagine you have a deck of cards, and you want to find the probability of drawing an Ace (event A)…