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 $$\dfrac{1}{6}$$.

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 $$\dfrac{1}{3}$$.

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

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.