Conditional probability

The conditional probability measures the probability of a certain event while knowing previous information about another event.

For example, if we want to calculate the probability that, after having thrown a dice, a $6$ comes out, we already know, by the rule of Laplace, that the probability is ${\displaystyle \frac{1}{6}}$.

Nevertheless, if we have the information that the result has been an even number, there are only three possibilities: $2,4$ and $6$, therefore the probability happens to be higher, of ${\displaystyle \frac{1}{3}}$.

Given two events $A$ and $B$, such that $P(B)\ne 0$, we call probability of $A$ conditioned on $B$, and we write $P(A/B)$, to: $$P(A/B)={\displaystyle \frac{P(A\cap B)}{P(B)}}$$

From the formula of the conditional probability we can derive an expression that will turn out to be very useful for us further on:

$$P(A\cap B)=P(A/B)\cdot P(B)$$

This expression is known as a principle of the compound probability.