A thief steals an ATM card and must randomly guess the correct

sixsix​-digit

pin code from a

44​-key

keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first​ try?

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Part 1

The number of possible codes is

enter your response here.

​(Type an integer or fraction. Simplify your​ answer.)

To calculate the number of possible codes, we need to consider two factors:

1. The thief is guessing a 6-digit pin code.
2. Each digit in the pin code can be selected from a keypad with 4 keys.

Since repetition of digits is allowed, each of the 6 digits in the pin can be independently selected from the 4 available keys.

So, the total number of possible codes is given by:

\[
\text{Total possible codes} = 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^6
\]

Let's calculate \( 4^6 \).

\[
4^6 = 4096
\]

Thus, the total number of possible codes is 4096.

### Probability of a correct guess on the first try:
Since there is only one correct pin code, the probability of guessing it correctly is:

\[
\text{Probability} = \frac{1}{4096}
\]

### Final answer:
- The number of possible codes is \( 4096 \).
- The probability of a correct guess on the first try is \( \frac{1}{4096} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What if the keypad had 10 keys instead of 4?
2. How does the probability change if the pin code is 4 digits long?
3. What if repetition of digits wasn't allowed?
4. How can this problem be related to a lottery system with similar guessing rules?
5. How would the number of possible codes change if the pin code was 8 digits long?

**Tip:** In probability problems, always carefully consider if repetition is allowed or not—it can significantly impact the total number of outcomes.

A thief steals an ATM card and must randomly guess the correct six-digit pin code from a 4-key keypad. Repetition of digits is allowed. What is the probability of a correct guess on the first try?

Learn how to calculate the probability of correctly guessing a 6-digit PIN code from a 4-key keypad with repetition allowed. This example involves basic probability and combinatorics, making it suitable for students in Grades 9-12. Includes step-by-step explanation and related concepts.

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