A thief steals an atm card and must randomly guess the correct four​-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 probability is 1/4096.

Since repetition is allowed, each digit could be any of the 8 on the keypad; this makes the number of choices 8*8*8*8 = 4096.  Only one of those is correct, so the probability is 1/4096.