The digits 1, 2, 3 and 4 are arranged at random, each used exactly once, to form a four-digit number. What is the probability that the number formed is NOT divisible by 4?

There are 4! = 24 such numbers. A number is divisible by 4 exactly when its last two digits form a multiple of 4, and from the digits 1, 2, 3, 4 without repetition the only such endings are 12, 24 and 32. Each ending leaves 2! = 2 ways to fill the first two places, so $3\times 2=6$ numbers are divisible by 4. Complementary counting gives 24 − 6 = 18 numbers, a probability of 18/24 = 3/4. Answering 1/4 stops at the divisible case.