Conditional Probability

Conditional probability describes how the probability of an event changes when we are given some additional information.

Given two events $A$ and $B$, the conditional probability of $A$ given $B$ is written $\mathbb{P}(A \mid B)$. It tells us the probability of $A$ under the assumption that $B$ has occurred.

The general formula is:

$$\mathbb{P}(A \mid B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$$

This is defined whenever $\mathbb{P}(B) \ne 0$.

For example, suppose we draw one card from a standard pack and are told that it is a face card: a Jack, Queen, or King. We’d now like to know the probability that the card is a King, taking this into account.

There are 12 face cards altogether, of which 4 are Kings. The conditional probability that the card is a King is:

$$\mathbb{P}(\text{King} \mid \text{Face card}) = \frac{4}{12} = \frac{1}{3}$$

The formula comes from the fact that conditioning changes the set of possibilities we are considering. Once we know that the card is a face card, the other 40 cards are no longer relevant to the calculation.