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The problem involves determining a specific value, \( c \), given the probability that the duration of a class exceeds \( c \) in a continuous uniform distribution between 49.2 and 55.5 minutes. We are given that \( P(x > c) = 0.323 \).

The formula for the probability in a continuous uniform distribution from \( a \) to \( b \) is:
\[ P(x > c) = \frac{b - c}{b - a} \]

Here, \( a = 49.2 \) and \( b = 55.5 \). Plugging in the values:
\[ 0.323 = \frac{55.5 - c}{55.5 - 49.2} \]

We can solve this equation to find \( c \).

Let's calculate that now.The duration \( c \) of the randomly selected class, such that the probability that the class lasts longer than \( c \) minutes is 0.323, is approximately 53.5 minutes when rounded to one decimal place.

Would you like to delve into how we solved this problem or any other details? Here are some related questions you might consider:

1. How does the uniform distribution differ from other probability distributions?
2. What would the probability \( P(x > c) \) be if \( c \) were set to the lower bound of the class duration range?
3. How would the solution change if the probability \( P(x > c) \) were greater than 0.5?
4. What is the expected value and standard deviation for the duration of this class?
5. Can you explain the significance of the \( a \) and \( b \) parameters in the uniform distribution?

**Tip:** When working with probabilities in continuous distributions, remember that the total area under the probability density function must equal 1. This principle helps in validating whether your calculations are correct.

The duration of a professor's class has a continuous uniform distribution between 49.2 minutes and 55.5 minutes. If one class is randomly selected and the probability that the duration of the class is longer than a certain number of minutes is 0.323, then find the duration of the randomly selected class.

Determine the duration of a class when given a uniform distribution between 49.2 and 55.5 minutes. Using probability theory, solve for the time c such that the probability P(x > c) equals 0.323. This is a great exercise in understanding continuous uniform distributions and probability.

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