“If we have an E.M.T. in the house, I think somebody got faint,” he said calmly when a woman keeled over in front of the stage in Cincinnati. “They just need a little water and some juice.”Unless this woman was a gender enigma like Pat from "Saturday Night Live," he should have said "she," not "they."
Somebody is a singular pronoun and needs to be paired with another singular pronoun. Sometimes, this is tough--particularly when you don't know the sex of the somebody in question.
There are also plenty of examples of great writers who've ignored this principle and use "their" as an all-purpose possessive pronoun. This doesn't make it right. For those of us who like a little precision with our speech and writing, keeping sentences parallel is the goal.
We don't love the "he or she" construction as a workaround. It's pretty darned stuffy. But this doesn't mean we shrug our shoulders and give up.
The better alternative is to use the plural when you really don't know the sex of the person in question. And when you do—let's say, when a woman has fainted at your glamorous feet—there's no need to further rob her of her dignity and neuter her.