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 AAA)…