Here's a place where the free market doesn't apply for the free market requires rational actors; there's nothing rational about getting hurt, passing out, and waking up in a hospital with a bill.
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Why would the free market not apply? What can be more rational then planning properly for emergencies that involve you living or dying? If anything it's irrational to hope that some government run "free" emergency care will help you out.
Lets take a typical Saturday night admission to an emergency room. Girl gets hammered and falls on her high heels and breaks her ankle.
Can a drunk person be expected to make rational plans? No.
Do we want to live in a society where a drunk woman with a broken ankle is supposed to look after her own care? No.
Under these circumstances, there can't be a market solution.
So in your example this woman made no prior emergency plans whatsoever via insurance and to top it off got drunk and the rest of society now should be forced to take care of her. We'll in that case why bother with any market solutions at all if people don't need to plan for their own lives. Talk about perverse incentives.