A thief steals an ATM card and must randomly guess the correct four-digit pin code from a 7 -key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first try? The number of possible codes is (Type an integer or fraction. Simplify your answer.)

To determine the probability of correctly guessing a four-digit PIN code from a 7-key keypad, we first need to calculate the total number of possible PIN codes. Since repetition of digits is allowed and there are 7 keys, each digit in the four-digit PIN can be any of the 7 keys. Therefore, the total number of possible PIN codes can be calculated as follows: \[ \text{Total possible codes} = 7 \times 7 \times 7 \times 7 = 7^4 \] Calculating \(7^4\): \[ 7^4 = 7 \times 7 \times 7 \times 7 = 49 \times 49 = 2401 \] So, there are 2401 possible four-digit PIN codes. Now, the probability of guessing the correct PIN code on the first try is the ratio of the number of successful outcomes (which is 1, since there is only one correct PIN) to the total number of possible outcomes (which is 2401). Thus, the probability \(P\) is given by: \[ P = \frac{\text{Number of successful outcomes}}{\text{Total possible codes}} = \frac{1}{2401} \] Therefore, the probability of a correct guess on the first try is: \[ \frac{1}{2401} \]