What is the chance of dental anaesthesia going into a blood vessel when administered?

It can happen: All practitioners who give injections are trained to "aspirated" or draw back on the syringe to be sure the needle hasn't entered a blood vessel. Sometimes the anesthetic gets into your bloodstream despite precautions. This is not dangerous. The vasoconstrictor in the anesthetic may give you a rapid pulse for a few minutes. By the time you're numb it has usually passed.