This is largely a factor of where the custody dispute takes place. Some jurisdictions favor mothers over fathers as much as 80% (although that number is decreasing as joint custody is becoming more politically and culturally popular nationwide). Other jurisdictions favor neither parent based upon the parent’s sex. To know what a father’s odds of winning sole or joint custody are, he would need to check the statistics in the state (and perhaps even the county) where his child custody dispute would be tried.
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