What are the chances of a child getting into the little leagues of a
any major sport?

Contact the club and ask about children's teams.
Most sports have teams for children.
Where are you and which sport are you talking about?

a bunch of teams have contact info right on their website - pending
they have one.  depending on what state/area you are from, this site
may help

http://www.sportsfansofamerica.com/LINKS/Baseball/LL/Main.htm

1st it depends on the sport itself, try joining the child in a club
for a sport in which he/she is interested. the coach will have links
for getting him/her into a junior league. All major sports teams have
junior teams as well....