You do have rights when buying privately, although fewer rights because certain parts of the Consumer Rights Act do not apply. For example, the car does not have to be in a satisfactory quality or fit for purpose.However, the seller must accurately describe the car, such as the number of previous owners. They must also not misrepresent it, for example not disclosing that it has been involved in an accident or providing a false service history.

During the first six months it then becomes the responsibility of the seller to prove that the problem wasn't there when they sold it, rather than it being up to the buyer to prove that is was there. If the individual refuses to accept your rejection, you will need to take legal action to reject the vehicle. However, this is expensive and there's no guarantee that you will win.