Question

a thief steals an atm card and must randomly guess the correct seven - digit pin code from a 5 - 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.) the probability that the correct code is given on the first try is . (type an integer or fraction. simplify your answer.)

Explanation:

Step1: Calculate number of possible codes

Since repetition of digits is allowed and there are 5 options for each of the 7 - digit pin code, by the multiplication principle, the number of possible codes is $5\times5\times5\times5\times5\times5\times5 = 5^7=78125$.

Step2: Calculate probability of correct guess on first try

The probability $P$ of a correct guess on the first try is the ratio of the number of favorable outcomes (1 correct code) to the number of total possible outcomes. So $P=\frac{1}{5^7}=\frac{1}{78125}$.

Answer:

The number of possible codes is $78125$
The probability that the correct code is given on the first try is $\frac{1}{78125}$