In California (Florida could be different), your insurance company would send the entire bill to the other insurance company. This process is called subrogation, making the responsible person (usually their insurance company) pay the cost of the damage.
Once your insurance company recieves a check from the other carrier, they will send you a check to cover your deductible. Not much work is needed on this beyond just giving it some time.
A few notes: Contact your agent, he will give you some insight on the process and should be able to find out if the claim is being subrogated (a ticket doesn't always prove fault).
Also, don't spend the check before you get it. It is usually a reasonably quick process, but sometimes drags on, and sometimes doesn't happen. The biggest issue is always the other person didn't really have insurance, in which case the carrier will not pay and you will have to wait for the person who hit you to pay.... unlikely to ever happen.
Again, you're agent can walk you right through this process.
Dan
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