A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 47.0 and 57.0 minutes. Find the probability that a given class period runs between 50.75 and 51.0 minutes. Find the probability of selecting a class that runs between 50.75 and 51.0 minutes. $$ \square $$ (Round to three decimal places as needed.)

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Let the length of a class period be the random variable $$ X $$ that is uniformly distributed on the interval $$[47.0, 57.0]$$. The probability density function (pdf) of a uniform distribution on an interval $$[a, b]$$ is given by

$$
f(x) = \frac{1}{b-a}, \quad \text{for } a \leq x \leq b.
$$

Here, $$ a = 47.0 $$ and $$ b = 57.0 $$, so the pdf becomes

$$
f(x) = \frac{1}{57.0 - 47.0} = \frac{1}{10}.
$$

We need to find the probability that $$ X $$ falls between $$ 50.75 $$ and $$ 51.0 $$ minutes. This probability is given by

$$
P(50.75 \leq X \leq 51.0) = \int_{50.75}^{51.0} \frac{1}{10} \, dx.
$$

Since the pdf is constant, the integral simplifies to

$$
P(50.75 \leq X \leq 51.0) = \frac{1}{10} (51.0 - 50.75) = \frac{1}{10} \times 0.25 = 0.025.
$$

Thus, the probability that a given class period runs between $$ 50.75 $$ and $$ 51.0 $$ minutes is

$$
\boxed{0.025}.
$$
The probability that a class runs between 50.75 and 51.0 minutes is 0.025.

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