Questions: 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 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 125000. (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.)

Solution Steps

To solve this problem, we need to determine the number of possible lock combinations and the probability of guessing the correct combination on the first try.

(a) To find the number of different three-number lock combinations, we need to consider that each of the three positions in the combination can be any of the 50 numbers, and repetitions are allowed. Therefore, the total number of combinations is \(50 \times 50 \times 50\).

(b) The probability of guessing the correct combination on the first try is the reciprocal of the total number of combinations.

1. Calculate the total number of combinations by raising the number of possible numbers (50) to the power of the number of positions (3).
2. Calculate the probability of guessing the correct combination by taking the reciprocal of the total number of combinations.

To find the total number of different three-number lock combinations, we use the formula:

\[ \text{Total Combinations} = n^{\text{positions}} = 50^3 = 125000 \]

The probability of guessing the correct lock combination on the first try is given by the reciprocal of the total number of combinations:

\[ \text{Probability} = \frac{1}{\text{Total Combinations}} = \frac{1}{125000} = 8 \times 10^{-6} \]

Final Answer