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.