To find the probability that a class period runs between 50.75 and 51.25 minutes, given that class lengths are uniformly distributed between 48.0 and 58.0 minutes, we can use the properties of the uniform distribution.

For a uniform distribution $U(a, b)$, where $a = 48.0$ and $b = 58.0$, the probability density function (PDF) is given by: $f(x) = \frac{1}{b - a} = \frac{1}{58.0 - 48.0} = \frac{1}{10} = 0.1$

Step 1: Determine the Interval Length

The probability of a value falling between two points $x_1$ and $x_2$ within a uniform distribution is: $P(x_1 \leq X \leq x_2) = f(x) \times (x_2 - x_1)$ In this case: $x_1 = 50.75 \quad \text{and} \quad x_2 = 51.25$ $x_2 - x_1 = 51.25 - 50.75 = 0.5$

Step 2: Calculate the Probability

Substitute into the formula: $P(50.75 \leq X \leq 51.25) = 0.1 \times 0.5 = 0.05$

Conclusion

The probability that a given class period runs between 50.75 and 51.25 minutes is 0.05, or 5%.

Would you like further details on uniform distribution or have any questions?

Here are some related questions you might find useful:

1. What is the probability of a class running exactly 55 minutes in a uniform distribution?
2. How do you find the mean of a uniform distribution?
3. What is the variance of a uniform distribution with bounds of 48.0 and 58.0?
4. How would the probability change if the distribution were not uniform?
5. What if the class length were between 50 and 54 minutes?

Tip: For uniform distributions, the probability over any interval can be found by multiplying the density by the interval length.