Counting and Probability 

If any two of the dice show an odd number, then this is impossible, so at most one of the dice can show an odd number. We take two cases:

Case 1: If exactly one of the dice shows an odd number, then all three other dice must show a multiple of 4; which can only be the number 4: The probability that this occurs is 4 × ½ × (1/6)³ = 1/108

Case 2: If all of the dice show even numbers, then the condition is satisfied. The probability that this occurs is (1/2)⁴ = 1/16

The total probability is: 1/108 + 1/16 = 31/432