Conditional Probability

probability-theory

Definition

Conditional Probability

Conditional probability is the probability of an event happening given that another event has already occurred. It’s the probability of an event given additional information.

The conditional probability of $A\in \mathfrak{S}$ given $B\in \mathfrak{S}$ is defined by:

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}\text{for}P(B)>0$$

Independence

If two events $A$ and $B$ are independent, i.e. $A\perp B$, and $P(B)>0$, then:

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

Therefore:

$$P(A\mid B)=P(A)⟺A\perp B$$