In a traditional family, father is preferred to be responsible with issues of property rights of the minor whereas the mother is favored with the issue of being a caretaker of the minor, but when there is a custody battle neither the father nor the mother are automatically given the custody.
Section 6 of the Hindu Minority and Guardianship Act, 1956 states that the natural guardian of a Hindu minor is the father and after him, the mother. It also states that where the minor has not completed the age of five years then the custody would be with the mother. Under section 6(b) in case of an illegitimate minor child it is the mother and after her, the father. But, under Section 13 of the same Act, the welfare of the child is for paramount consideration by the Court while deciding custody. So, all these provisions do not stand any value if they are in conflict with welfare of child. Both father and mother can be granted custody.
Muslim law provides that mother is entitled to the custody of her male child until he has completed the age of seven years and of the female child until she has attained puberty (i.e. at the age of 14or 15 years generally) unless found guilty of misconduct. Section 17 of Guardian and Ward Act provides that custody is to be decided as per the personal laws. Section 7 and 17 of the Guardian and Ward Act also provides the welfare of child as main consideration. So the Court tries to maintain a balance between the specific directions of the personal laws and welfare of child.
Now, coming on to the point why people have a misconception that women are more likely to win the battle of custody. Well, the law generally treats mother and father equally while deciding custody. But sometimes the mothers are preferred for the custody like in case of girl child as it is normally considered that the girls require guidance of their mother. Also merely because the mother is working lady she should not be refused custody of the female child as held in Thrity Hoshie Dolikuka vs. Hoshiam Shavaksha Dolikuka B, 1982 AIR 1276, 1983 SCR (1) 49 on 4 August, 1982. So in cases of custody of infant child and girl child, the Court considers the best interest of child with mother and prefers them over father but it is not necessary if the facts and circumstances of the case are indicating something contrary to this fact.