The lengths of a professor’s classes has a continuous uniform distribution between 50.0 min and 52.0 min.  If one such class is randomly selected, find the probability that the class length is less than 50.5 min.


For a continuous uniform distribution, the probability density function \( f(x) \) is constant over the interval \([a, b]\), where \(a\) and \(b\) are the minimum and maximum values. In this case:

- Minimum class length \( a = 50.0 \) minutes
- Maximum class length \( b = 52.0 \) minutes

The probability of an event in a continuous uniform distribution over an interval \([a, b]\) is given by:
\[
P(X < x) = \frac{x - a}{b - a}
\]

Here, we are asked to find the probability that the class length is less than \( 50.5 \) minutes.

1. Substitute the values into the formula:
   \[
   P(X < 50.5) = \frac{50.5 - 50.0}{52.0 - 50.0}
   \]

2. Simplify:
   \[
   P(X < 50.5) = \frac{0.5}{2.0} = 0.25
   \]

### Answer:
The probability that the class length is less than 50.5 minutes is **0.25** or **25%**.

Would you like further details or have any questions?

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1. How would the probability change if the class length interval was 48.0 to 52.0 minutes?
2. What is the probability of selecting a class longer than 51.0 minutes?
3. How does a continuous uniform distribution differ from a discrete uniform distribution?
4. How can this concept apply to real-world scheduling issues?
5. What if the problem asked for the probability of a class length exactly 51 minutes?

**Tip:** In continuous uniform distributions, probabilities of exact values (like \(P(X = x)\)) are always zero because the probability density function is spread over an interval.

The lengths of a professor’s classes has a continuous uniform distribution between 50.0 min and 52.0 min. If one such class is randomly selected, find the probability that the class length is less than 50.5 min.

Explore how to calculate the probability that a professor’s class length is less than 50.5 minutes, given a continuous uniform distribution between 50.0 and 52.0 minutes. This solution includes the key formula P(X < x) = (x - a) / (b - a) and is suitable for students in Grades 10-12.

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