Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are known. I notice that in particular, if the bundle is trivial, then $E=B\times F$ and $w(E)=w(B)w(F)$.

**Question:** In general, are there any formulas to compute the Stiefel-Whitney class of the total space $E$
$$
w(E)
$$
in terms of $w(B)$ and $w(F)$? Or even in terms of the cohomology ring
$$
H^*(B;\mathbb{Z}_2), H^*(F;\mathbb{Z}_2)
$$
And other factors?