A certain three-cylinder combination lock has 75 numbers on it. To open it, you turn to a number on the first cylinder, then to a second number on the second cylinder, and then to a third number on the third cylinder and so on until a three-number lock combination has been effected. Repetitions are allowed, and any of the 75 numbers can be used at each step to form the combination. (a) How many different lock combinations are there? (b) What is the probability of guessing a lock combination on the first try? (a) The number of different three-number lock combinations is 421875 (Type an integer or fraction. Simplify your answer.) (b) The probability that the correct lock combination is guessed on the first try is (Type an integer or fraction. Simplify your answer.)

Step 1 ：For part (a), since there are 75 numbers on each cylinder and there are 3 cylinders, the total number of combinations would be \(75 \times 75 \times 75\). This is because for each number on the first cylinder, there are 75 possibilities on the second cylinder, and for each of those, there are 75 possibilities on the third cylinder.

Step 2 ：Calculating the total number of combinations gives us \(75 \times 75 \times 75 = 421875\).

Step 3 ：For part (b), the probability of guessing the correct combination on the first try would be 1 divided by the total number of combinations. This is because there is only one correct combination out of all the possible combinations.

Step 4 ：Calculating the probability gives us \(\frac{1}{421875} \approx 2.37 \times 10^{-6}\).

Step 5 ：Final Answer: (a) The number of different three-number lock combinations is \(\boxed{421875}\)

Step 6 ：(b) The probability that the correct lock combination is guessed on the first try is approximately \(\boxed{2.37 \times 10^{-6}}\)