For two events $A$ and $B$, the conditional probability of event $A$, is the probability of event $A$ given the occurence (or non-occurence) of event $B$.

The conditional probability of $A$ given $B$, written as $\mathbb{P}\left(A\mid B\right)$, is defined by

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

If events $A$ and $B$ are independent, $\mathbb{P}\left(A\cap B\right)=\mathbb{P}\left(A\right)×\mathbb{P}\left(B\right)$ . Hence

$\mathbb{P}\left(A\mid B\right)=\mathbb{P}\left(A\right)$ $\mathbb{P}\left(B\mid A\right)=\mathbb{P}\left(B\right)$

In other words, the probability of $A$ given $B$ is just the probability of $A$ (regardless of $B$ — whether $B$ has occured or not doesn't affect the probability of $A$, because they are independent events).

Examples:

• In a certain region, on any day of the year, the probability that it's cloudy is 0.4. It's also known that there's a 0.3 probability that any day is a cloudy and rainy day. Given that today is cloudy, what is the probability that it will rain?

Let $C$ be the event that it's cloudy and $R$ be the event that it rains.

$\begin{array}{rl}\mathbb{P}\left(C\right)& =0.4\\ \mathbb{P}\left(R\cap C\right)& =0.3\\ \mathbb{P}\left(R\mid C\right)& =\frac{\mathbb{P}\left(R\cap C\right)}{\mathbb{P}\left(C\right)}\\ & =\frac{0.3}{0.4}\\ & =0.75\end{array}$

Given today is cloudy, there's a 75% chance it will rain today.

• In a city, the ratio between men and women is 6:4. Thirty percent of the men are vegetarian. What percentage of the city residents are vegetarian men?

Let $M$ be the probability of any random resident being a man and $V$ be the probability that any random resident being a vegetarian.

$\begin{array}{rl}\mathbb{P}\left(M\right)& =0.6\\ \mathbb{P}\left(V\mid M\right)& =0.3\\ \mathbb{P}\left(V\mid M\right)& =\frac{\mathbb{P}\left(V\cap M\right)}{\mathbb{P}\left(M\right)}\\ 0.3& =\frac{\mathbb{P}\left(V\cap M\right)}{0.6}\\ \mathbb{P}\left(V\cap M\right)& =0.3×0.6\\ \mathbb{P}\left(V\cap M\right)& =0.18\end{array}$

Eighteen percent of the city residents are vegetarian men.