Question

a thief steals an atm card and must randomly guess the correct five - 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

Since there are 7 - key keypad and repetition is allowed, for each of the 5 - digit pin code, there are 7 possibilities.

Step2: Calculate total number of possible codes

By the multiplication principle, the total number of possible 5 - digit pin codes is $7\times7\times7\times7\times7 = 7^5$.
$7^5=7\times7\times7\times7\times7 = 16807$.

Step3: Calculate probability of correct guess

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. So the probability is $\frac{1}{7^5}=\frac{1}{16807}$.

Answer:

$\frac{1}{16807}$