To determine the probability that a randomly generated 10-character password contains all letters, we need to consider the set of characters and the length of the password.
Here’s a step-by-step explanation of how we can solve the problem:
### Step-by-Step Solution:
1. Identify the Components:
- Total characters available (letters + digits): 26 letters + 10 digits = 36 characters.
- Total letters only: 26 letters.
- Password length: 10 characters.
2. Calculate the Probability:
- For each character in the password, there is a probability that it could be a letter.
- Since the selection of each character is independent, the probability of the entire password being composed of letters is the probability of selecting a letter raised to the power of the length of the password.
3. Find the Probability of One Character Being a Letter:
[tex]\[ \frac{\text{Number of letters}}{\text{Total number of characters}} = \frac{26}{36} \][/tex]
4. Extend This Probability for All Characters in the Password:
We raise the probability of one character being a letter to the power of the password length:
[tex]\[ \left(\frac{26}{36}\right)^{10} \][/tex]
5. Calculate the Numerical Value:
Using the given numerical values, the probability calculation has already given a result:
[tex]\[ \left(\frac{26}{36}\right)^{10} \approx 0.03861077083162305 \][/tex]
6. Round the Result to Four Decimal Places:
The result, rounded to four decimal places, is:
[tex]\[ 0.0386 \][/tex]
### Conclusion:
The probability that a randomly generated 10-character password will contain all letters is approximately [tex]\(0.0386\)[/tex] when rounded to four decimal places.
