Yes. An uninsured driver can file a claim against the insured's driver policy and seek personal injury/property damage compensation. However, the uninsured driver must proove the insured driver was at fault for the accident.
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An uninsured motorist can make a claim against an insured motorist who is at fault in an accident. Also, a motorist that carries uninsured motorist coverage can make a claim against their own insurance company when they are hit by an uninsured driver.