The probability that he will guess an answer right is 1/4.
The probability that he will guess an answer wrong is 3/4.

This is a cumulative binomial probability where the number of trials 
is n=6, with the probability of a success in 1 trial is 1/4.

With a TI-83 or 84,  it's 1-binomcdf(6,1/4,2). Ans: 0.1694335937

Or without a calculator that has binomial probability feature:

Case 1: Guessing exactly 3 correct:

Choosing exactly 3 to get correct = Choosing exactly
3 to get wrong = 6C3 = (6*5*4)/(3*2*1) = 20 ways.
Guess the wrong answer for the first wrong one 3 ways.
Guess the wrong answer for the second wrong one 3 ways.
Guess the wrong answer for the third wrong one 3 ways.
That's 20×3×3×3 = 540 ways

Case 2: Guessing exactly 4 correct:

Choosing exactly 4 to get correct = Choosing exactly
2 to get wrong = 6C2 = (6*5)/(2*1) = 15 ways.
Guess the wrong answer for the first wrong one 3 ways.
Guess the wrong answer for the second wrong one 3 ways.
That's 15×3×3 = 135 ways

Case 3: Guessing exactly 5 correct:

Choosing exactly 5 to get correct = Choosing exactly
1 wrong = 6C1 = 6 ways.
Guess the wrong answer for the wrong one 3 ways.
That's 6×3 = 18 ways

Case 4: Guessing all 6 correct
There is only 1 way to guess them all correct

Total number of ways to pass: 540+135+18+1 = 694 ways to pass

Total number of ways to answer:

(4 ways to answer question 1) times
(4 ways to answer question 2) times
(4 ways to answer question 3) times
(4 ways to answer question 4) times
(4 ways to answer question 5) times
(4 ways to answer question 6) equals 4×4×4×4×4×4 = 46 = 4096

 
  
Edwin