A thief steals an ATM card and must randomly guess the correct five-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___.
The correct answer and explanation is :
The number of possible codes is 5⁵ = 3125 because the thief can choose each of the 5 digits independently, with 5 options (0–4, assuming a 5-key keypad).
Probability Calculation:
Since only one of these 3125 codes is correct, the probability of guessing the correct PIN on the first try is:
[
P = \frac{1}{3125}
]
Explanation:
- Understanding the Problem
The ATM card requires a 5-digit PIN, with each digit selected from a keypad with only 5 keys. Since repetition is allowed, each digit can independently take any of the 5 values. - Total Number of Possible Codes
Since each of the 5 positions can be filled by any of the 5 digits, the total number of unique PIN codes is calculated using the formula for permutations with replacement:
[
5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125.
] - Probability of Correct Guess
Since only one out of these 3125 possibilities is the correct PIN, the chance of guessing it correctly in one attempt is:
[
\frac{1}{3125} \approx 0.00032 \text{ or } 0.032\%.
]
This means that the probability of failure is extremely high (99.968%), making it almost impossible for the thief to succeed. - Conclusion
This problem highlights how PIN security relies on combinatorial possibilities. Even with only 5 keys, the sheer number of potential PINs makes it highly improbable for an unauthorized user to guess the correct one in a single attempt.
