P(6 less than or equal to X less than or equal to 14)

To calculate \( P(6 \leq X \leq 14) \), we need to know the context of \( X \). Typically, this would be the probability of a random variable \( X \) taking values between 6 and 14 inclusive. The calculation depends on the probability distribution of \( X \). 

Here are the possible steps depending on the distribution:

### 1. **If \( X \) follows a continuous distribution (e.g., normal distribution):**
The probability is found by integrating the probability density function (PDF) of \( X \) from 6 to 14. 

### 2. **If \( X \) follows a discrete distribution:**
The probability is the sum of the probabilities of each value \( X \) can take between 6 and 14.

### Example for Normal Distribution:
If \( X \) is normally distributed as \( X \sim N(\mu, \sigma^2) \) (with mean \( \mu \) and standard deviation \( \sigma \)), then the probability \( P(6 \leq X \leq 14) \) is:

\[
P(6 \leq X \leq 14) = P\left( Z \leq \frac{14 - \mu}{\sigma} \right) - P\left( Z \leq \frac{6 - \mu}{\sigma} \right)
\]
where \( Z \) is the standard normal variable.

Do you have information about the distribution or values for \( \mu \) and \( \sigma \) for this case?

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Would you like further details or have any questions? Here are 5 related questions to expand on this topic:

1. What is the probability distribution of \( X \) in this problem?
2. If \( X \) is normally distributed, do you have the mean (\( \mu \)) and standard deviation (\( \sigma \)) values?
3. Is \( X \) a discrete or continuous random variable?
4. Do you need help calculating the cumulative probability for a specific distribution?
5. Would you like me to walk you through the normal distribution tables?

**Tip**: For normal distributions, always convert to the standard normal variable \( Z \) using \( Z = \frac{X - \mu}{\sigma} \) for easier calculations using tables or software.

Understand how to calculate the probability P(6 ≤ X ≤ 14) for both continuous and discrete random variables. This guide explores key concepts like probability distributions, including normal distributions, and provides step-by-step solutions with relevant formulas and theorems. Suitable for advanced high school and college-level students.

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