He needs 6 correct answers to pass and he already has 4. That means he needs 2 more to pass. It's easier to look at how he can fail. That means no correct answers or only 1 correct answer. The probabilities are:
Probability of no correct answers:
8 choose 0 (1/2)^8
Probability of 1 correct answer:
8 choose 1 (1/2)^8
Add these 2 up, subtract the result from 1 and you have the probability he has 2 or more correct answers.