Question

a thief steals an atm card and must randomly guess the correct five - digit pin code from a 8 - 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 each digit of the 5 - digit pin code can be chosen in 8 ways (8 - key keypad) and repetition is allowed, by the multiplication principle, the number of possible codes is $8\times8\times8\times8\times8=8^5 = 32768$.

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 $P=\frac{1}{8^5}=\frac{1}{32768}$.

Answer:

The number of possible codes is $32768$.
The probability that the correct code is given on the first try is $\frac{1}{32768}$.