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)…