What is the likelihood of a random person having a sister?

All that you need to know, really, is how many people in general have sisters.  Just measure a random population and derive the answer from that.

You could do a summation using the proportion of every single case (children with no siblings, children with one sibling, children with two siblings, etc.), but you can't say what proportion of the population each particular case represents without some sort of measurement or sampling, so why beat around the bush and sample indirectly? 

As to the infinity of cases problem, random odds (assuming it applies) would impose a family size where the chances of no female children becomes trivial and can be ignored, so there is a family size where you can say that all the children will have a female sibling (not only one girl, but two, out of the children).

You cannot do a global probability without knowing the proportion of instances represented by each case.  Since you have to already do a study to learn that, why not just deal with the question directly?

You'd have to know the distribution of the numbers of children in families with at least one child.  Then you could estimate the probability that a given individual is from a family of n children.  Multiply that by the probability that an individual n-1 siblings has at least one female siblings.

The usual assumption is that any child is equally likely to be a male or female, but that doesn't match your "at least 1 of them was female" condition.  If that condition were true (along with 2 children in every family) then every male would have a sister with probability 1.  That's easy.  The probability that a randomly chosen female has a sister is more interesting. 

Suppose there are F families, each with 2 children, and M families have one male and one female child.  Then F-M of the families have 2 female children.  So M of the female children are in families with a male child and have no sister.  The remaining 2(F-M) female children have sisters.  Out of a total of M + 2(F-M) = 2F - M female children, 2F - 2M of them have exactly one sister.  So the probability that a female child has a sister is (2F - 2M)/(2F - M).

But the stats are simpler for a randomly chosen child.  There are 2F children in all, and 2(F-M) are female children with sisters.  The probability that a person from that group has a sister is 2(F-M)/(2F) =(F-M)/F = 1 - M/F.  (Remember that F is families, not females.)