Correctness of the Betweenness centrality formula

Question

Betweenness centrality is defined as the number of shortest paths that go through a node in the graph.The formula is:

$$\sum_{s \neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$$

Where $\sigma_{st}$ is the total number of shortest paths from node $s$ to node $t$ and $\sigma _{st}(v)$ is the number of those paths that pass through $v$.

However it doesn't seem to me that the formula calculates what is defined. Why do we divide by the total number of shortest paths between $s$ and $t$ each time? Shouldn't we just divide by $2$ to compensate the fact that $s$ and $t$ will appear twice in different orders?