Final answer:
Option 1 contains a comma splice, joining two independent clauses with only a comma. This is a grammatical error which can be fixed by adding a coordinating conjunction, or by using a period or semicolon.
Step-by-step explanation:
The grammatically incorrect option that includes a comma splice is the first one. It reads: "Chicago is a vibrant and well-known American city, it is also known for having a lot of violent shootings and what some claim to be a corrupt and brutal police force." Here, two independent clauses are incorrectly joined by a comma. To correct a comma splice, you can use methods such as adding a coordinating conjunction, replacing the comma with a period or semicolon, or making one clause subordinate to the other.