Question

a thief steals an atm card and must randomly guess the correct three - 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.)

Explanation:

Step1: Calculate number of possibilities per digit

Each digit of the 3 - digit pin can be chosen in 7 ways since there are 7 keys on the keypad and repetition is allowed.

Step2: Use multiplication principle

By the multiplication principle of counting, the total number of possible 3 - digit codes is the product of the number of choices for each digit. So the number of possible codes is $7\times7\times7=7^{3}=343$.

Step3: Calculate probability

The probability of a correct guess on the first try is the number of favorable outcomes (1 correct code) divided by the number of total possible outcomes. The probability $P=\frac{1}{343}$.

Answer:

The number of possible codes is $343$.
The probability of a correct guess on the first try is $\frac{1}{343}$.