Kiara sets up a passcode on her smart phone, which allows only four-digit codes. A spy sneaks a look at Kiara's smart phone and sees her fingerprints on the screen over four numbers. What is the probability the spy is able to unlock the smart phone on his first try? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.


Since the spy knows that Kiara's passcode is a four-digit code and can only consist of the four digits she left fingerprints on, there are a limited number of possible combinations.

1. **Calculate the Total Combinations:**  
   With four distinct digits to choose from, the number of possible four-digit passcodes is a permutation of 4 digits taken 4 at a time. This can be calculated as:
   \[
   4! = 4 \times 3 \times 2 \times 1 = 24
   \]
   So, there are 24 possible four-digit codes.

2. **Determine Probability of Success on First Try:**  
   Since only one of these combinations is correct, the probability that the spy guesses the correct code on the first try is:
   \[
   \frac{1}{24}
   \]
   Converting this to a decimal gives:
   \[
   \frac{1}{24} \approx 0.041667
   \]
   Rounded to the nearest millionth, the probability is:
   \[
   0.041667
   \]

Thus, the probability that the spy unlocks the smartphone on the first try is \( \frac{1}{24} \) or \( 0.041667 \). 

Would you like further clarification on any step, or do you have any questions?

Here are five related questions you might find interesting:
1. What if Kiara's passcode were a six-digit code using only four distinct digits?
2. How does the probability change if the spy knew three digits but not their order?
3. What if the passcode allowed repeated digits? How would that affect the probability?
4. How would the scenario differ if the spy could make multiple attempts?
5. What is the probability of guessing a 4-digit passcode without any prior information about the digits?

**Tip:** When calculating probabilities involving permutations, consider how the order and number of items affect the total possible outcomes.

Kiara sets up a passcode on her smartphone, which allows only four-digit codes. A spy sneaks a look at Kiara's smartphone and sees her fingerprints on the screen over four numbers. What is the probability the spy is able to unlock the smartphone on his first try? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

Explore the probability of a spy guessing the correct four-digit smartphone passcode on the first attempt. This problem involves key probability and permutation concepts, with a detailed solution showing the calculation of total combinations and success probability, suitable for Grades 9-10.

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