What percent of women win child custody cases?

That's interpretive as there are four levels of custody.

Sole Custody

Joint Legal with Primary Residential

Joint Physical Custody

Bird Nest Custody

Fathers with primary or sole custody runs 15%.

Joint physical custody runs 25%

Mother with primary or sole custody is 60%

But, single mothers have sole custody by default in 100% of the cases until ruled on otherwise by the courts. Also, attorneys tell fathers not to try for custody as they don't win, and most attorneys that do try are not fully capable of representing fathers, so the mothers win.

In the cases where fathers do challenge for custody, 60% of them will be accused of child sexual abuse as a tactic by the mother to prevent him getting it, but even when successful, they can still lose as 30% of them learn they are not the father of one or more of the children. If he does win custody, he can than be accused of domestic violence committed some time in the previous 12 months, as which point an injunction stops him from getting the child until he can prove himself innocent. If he does prove himself innocent, or it was found that he defended himself when she attacked him, then he cannot have custody because that's still considered domestic violence against the mother.