When working with probabilities previously, we would have assumed that we are asked to state the probability of an event given that we are working with the entire universal set, $U$. Thus, for an event $A$, the probability is: $$ \begin{equation}\begin{aligned} P(A)=\frac{n(A)}{n(U)}\\ \end{aligned}\end{equation} $$
We can think of this as the conditional probability of $A$ given $U$: $$ \begin{equation}\begin{aligned} P(A|U)=\frac{n(A)}{n(U)}\\ \end{aligned}\end{equation} $$
What if we are given only the elements of the set $B$? The conditional probability of $A$ given $B$ would thus be: $$ \begin{equation}\begin{aligned} P(A|B)=\frac{n(A\cap B)}{n(B)}\\ \end{aligned}\end{equation} $$
This conditional probability is therefore the ratio of the elements in $A$ which are also in $B$ to the elements in $B$. This can be written as probabilities: $$ \begin{equation}\begin{aligned} P(A|B)=\frac{P(A\cap B)}{P(B)}\\ \end{aligned}\end{equation} $$