The results are similar for several studies that do not look just at fathers who seek sole custody. See this meta-analysis of several studies: http://www.amptoons.com/blog/files/Massachusetts_Gender_Bias....

I'll quote the relevant part:

"""
Although perceptions of bias that discourage fathers from seeking custody are a concern, n52 the outcome of cases in which custody is contested provides a more direct source of information about possible judicial gender bias. We heard testimony from George Kelly, a representative of Concerned Fathers, that in contested custody cases, mothers are awarded physical custody over 90% of the time. Mr. Kelly was unable to provide substantiation, however, n53 and our own investigation revealed a very different picture.

The statewide sample of attorneys who responded to the family law survey had collectively represented fathers seeking custody in over 2,100 cases in the last 5 years. n54 They reported that the fathers obtained primary physical custody in 29% of the cases, and joint physical custody in an additional 65% of the cases. Thus, when fathers actively sought physical custody, mothers obtained primary physical custody in only 7% of cases. The attorneys reported that the fathers had been primary caretakers in 29% of the cases in which they had sought custody.

The preliminary findings of the Middlesex Divorce Research Group relitigation study show a similarly high rate of paternal success, but fewer awards of joint physical custody. In their sample of 700 cases in Middlesex County between 1978 and 1984, fathers had sought custody in 57 cases (8.14% of the sample). In two-thirds of the cases in which fathers sought custody, they received primary physical custody (42% in which fathers were awarded sole legal and sole physical custody, plus [*832] 25% in which fathers were awarded joint legal and primary physical custody). Joint physical and joint legal custody was awarded in 3.5% of cases. In 11% of the cases, mothers received primary physical and joint legal custody; in 12%, mothers were awarded sole legal and physical custody; other custodial arrangements were ordered in the remaining cases. Thus, when fathers sought custody, mothers received primary physical custody in fewer than one-quarter of the cases in the Middlesex study. Information about which parent had been the primary caretaker was not available for the Middlesex cases.

These trends were apparent in an earlier study of a sample of 500 Middlesex County cases filed between 1978 and 1981. Fathers had sought sole custody in about 8% of the cases. They received sole custody in 41% of those cases, and joint custody in 38%. In 5% of the cases, custody went to someone other than a parent. In instances in which fathers sought sole custody, mothers received sole custody in only 15% of the cases (Phear et al., 1983).

These statistics may be a surprise to many. They are, however, consistent with findings in other states. A study of court records in Los Angeles County, California, in 1977 found that fathers who sought sole custody obtained it in 63% of the cases (up from a success rate of 37% in 1972) (Weitzman, 1985, p. 233). A nationwide survey of all reported appellate decisions in child custody cases in 1982 found that fathers obtained custody in 51% of the cases, up from an estimated 10% in 1980 (Atkinson, 1984).

The high success rate of fathers does not by itself establish gender bias against women. Additional evidence, however, indicates that women may be less able to afford the lawyers and experts needed in contested custody cases (see "Family Law Overview") and that, in contested cases, different and stricter standards are applied to mothers.
"""