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This problem involves calculating the probability that a normally distributed variable is less than a certain value.

Given:
- The mean \( \mu = 12.4 \) inches
- The standard deviation \( \sigma = 4.1 \) inches
- We need to find the probability \( P(X < 10) \) for a normally distributed random variable \( X \).

### Solution
1. **Standardize the value of 10** to a z-score, which is calculated by:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   Substituting in the values:
   \[
   Z = \frac{10 - 12.4}{4.1} = \frac{-2.4}{4.1} \approx -0.585
   \]

2. **Find the probability** for \( Z < -0.585 \) using the standard normal distribution table or a calculator.

   Looking up \( Z = -0.585 \) in the standard normal table or using a calculator gives approximately:
   \[
   P(Z < -0.585) \approx 0.2794
   \]

### Conclusion
The probability that a randomly selected item is less than 10 inches long is approximately **0.2794** or **27.94%**.

Would you like further explanation on this process or have any questions?

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Here are some related questions that might deepen your understanding:

1. How do we interpret a z-score in a normal distribution?
2. How would the probability change if the standard deviation were smaller?
3. What if we wanted the probability that the item is more than 10 inches long?
4. How can we use this process for values other than 10 inches?
5. How does a z-score help in comparing different data points within a normal distribution?

**Tip:** Always check if the values provided require a one-tailed or two-tailed probability, as it affects the interpretation and calculation of results.

A manufacturer knows that their items have a normally distributed length, with a mean of 12.4 inches, and standard deviation of 4.1 inches. If one item is chosen at random, what is the probability that it is less than 10 inches long?

Find the probability that a normally distributed item with a mean length of 12.4 inches and a standard deviation of 4.1 inches is less than 10 inches long. This solution covers calculating Z-scores and using the normal distribution table. Suitable for students in Grades 10-12.

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