Conditional Probability Changes the Universe

Conditional probability answers a probability question after restricting attention to an event that is already known to have occurred.

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

The denominator is the key idea. Once B is given, B becomes the active sample universe. This is why P(A|B) and P(B|A) generally differ: they use the same intersection but different denominators.

In a two-way table, highlight the conditioning row or column before calculating. In a tree, read a branch probability relative to its parent state rather than to the original root.

The Conditional Probability lab lets you switch the conditioning event while keeping the same canonical counts.